The nativity stories in Matthew and Luke are an example of what is known as a doublet - two different stories about the same event. As such, we should not attempt to harmonise the two stories, but read each on its own and on its own merits.

A careful reading of Luke 2:22,39 shows Jesus was taken via the Temple in Jerusalem direct to Nazareth, the home town of Joseph and Mary, soon after his birth. A careful reading of Matthew's Gospel shows that Bethlehem was the home town of Joseph and Mary, and they remained there until the arrival of the wise men, after which they were told to flee to Egypt. After the death of Herod, they began the return journey to Bethlehem in Judea but, being warned in a dream, turned aside and instead travelled to Galilee, where they settled in a city called Nazareth, thereby fulfilling the prophecy that Jesus be called a Nazarene. As far as Matthew is concerned, this is the first time Nazareth enters the picture.

I think the following selection is a fair representation of scholarly opinion on why it is not possible to harmonise the two gospel accounts, and why there is no meaning in trying to establish which event occurred first:

1. John Shelby Spong say (Born of a Woman: A Bishop Rethinks the Birth of Jesus, pages 89-90) that among people he knows in New Testament circles, the universal assumption is that the magi [wise men] were not actual people. Matthew was clearly writing Christian midrash. Spong goes on to say that Matthew is drawing a parallel between Jesus and Moses in the Old Testament: Joseph took his family to Egypt just as the patriarch Joseph had done, in order to escape death. The killing of all male babies reflects the story of the pharaoh ordering all the male babies under two to be killed.
2. Raymond E. Brown says, in An Introduction to the New Testament, page 114 that Luke's infancy narrative is not only massively different from Matthew's, but also in details is virtually irreconcilable with it.
3. Ian Wilson,in Jesus, page 47, discounts the nativity account in Luke's Gospel, saying that the author may have been trying to make it appear that he knew more about Jesus’ birth than he actually did.
4. Uta Ranke-Heinemann says in Putting Away Childish Things, page 11, Luke handles the facts arbitrarily, so that the facts themselves refute him.
5. Burton L. Mack says in Who Wrote the New Testament, pages 165-6, the stories of Jesus' birth and infancy are seen by biblical scholars as incredible, implausible, far-fetched fictions.

I would like if you at least read these first three sentences before you decide not to read it if you decide that. I looked into this as well and it was neat that you are too, also because I came across the very same question. But after further investigation, I have more reason to believe the wise men came to Jesus after Jesus being dedicated to the Lord. Here is why: The wise men took a while before coming to Jerusalem. The wise men come in "the" house. Joseph and Mary's town was Galilee. The wise men went to Bethlehem, but at night saw the star again. Herod found out his plan didn't come together. For much more detail and reason, read on:

The wise men said they come from the east. Now since they came to Jerusalem, I am sure Luke wasn't referring to the other side of the dead sea, but rather farther than that. And I found that apart from the other side of the dead sea, the next great water source to the east would of been about 500 miles from Jerusalem. So this would be at least a seven day journey. If they walked 3 mph the whole way there for a strait pace with no sleep with a strait path with no mountains, then they would get there in just under seven days. Obviously they had horses and there were mountains and curved paths and they slept. So with this in mind, these wise men I am sure if my research is right, were ones that thought the stars told the future and studies the stars. Now it is clear they saw the star once or soon after Jesus was born. So if they went strait away to Jerusalem, I think it can be safe to say it was after the circumcision of Jesus that they saw Jesus. But If they did not go strait away, why would that be? There are a few reasons of probability and possibility. 1st, For at least a seven day journey, they would have needed to pack for it. So this would of taken at least probably a day. It might of taken longer if they went into their city to get some supplies. Now before they packed, they would of needed to know that was what the star meant to them. So unless they studied it to know the star would appear at that time, then they would have gone early, which we have reason to believe didn't happen. So at least they stayed to see if the star appeared. They may have been studying the stars, but not this one. But when it appeared, it caught there attention and studied that consolation and star. So I do not know how long that would take, but lets say it took at least a day and at most a bit over a year and a half. Now also I am sure they would want to be sure that their interpretation of the star was correct before spending news, because if they were wrong, they would of given all of Israel a false hope of the Christ being on earth. Also we know at least two wise men come, so they would have to of agreed to go at a certain time. Who knows what exactly happened, but here is more reason of mine. I think it is safe to assume Joseph and Mary were in Bethlehem until they went to Jerusalem to dedicate Jesus to the Lord. So I believe there is no gap between Luke 38 and 39, but rather at 39 and technically 41. For the reasons I think they were at Nazareth when the wise men came to them was because of these things: It says the wise men came into their house. When Jesus was born, they were more in a small cave. Also it would seem that knowing they would go up on the 33 day of purification, they would decide to go back home. Because Luke says they went back to both of theirs own towns, which is the same town, Nazareth. We know Joseph had a house in his own town, so they probably didn't buy another house, but rather stayed in the inn when they could, before they returned to Nazareth. Because it would be packed when they got there because the Census was the counting of people, if my information is correct. So after the counting, they could have stayed in the hotel. Yes Herod sent them to Bethlehem, and to Bethlehem, they most probably went. Because if he was born in Bethlehem, and the wise men got there a while after the birth of Jesus, then Herod would have known they might not be there anymore. So he sent them to Bethlehem to find Jesus. Keep in mind, it was probably day when they went to Bethlehem, and they looked there and probably asked around, and then at night, the star appeared, and the path was set. Then they went to Nazareth and saw Jesus, and then it probably the night after the wise men saw Jesus that Joseph was warned to go to Egypt. So then that is that. The reason I have a problem with the other possibility is because the time would be really compacted. For after being born, it would probably be at least after Jesus was circumcised that the wise men came to Him. So that leaves 32 days. then after the wise men saw them, they left and Joseph and Mary, with Jesus, fled to Egypt. So this would be at least a 3 day journey. So let's just say at most 29 days are left. Herod finding out the wise men didn't do what he said and Herod killing all those babies would take at least a full day. So 28 days. the call back to Israel, 3 more days at least, so 25 days, Now Joseph is afraid of the new king, they now go to Nazareth. I heard that was at least a 2 day journey. 23 days. Now we have about 20 days to play around with. For they at least were at Nazareth for 3 days before eJoseph went back to Jerusalem. So this would be a lot of travel for a person just giving birth with only 20 days of rest in between. They probably might of had a donkey to ride, but who knows. Anyway, that is why I do not think that happened that way, but it is possible, especially if they didn't have to go to the dedication of Jesus only the day of, but anytime after the purification, which I think is wrong to think. Now last reasons for the first example to be right. Herod killed all the boys two years or younger. When Simon said something to Mary, he says Jesus is appointed for the falling and rising of many in Israel, and for a sign spoken against. So this falling and rising is not only of physical things, but more also of spiritual things. So after the first glimpse of that happened with the killing of the babies, Mary may have known that the sword piercing through her soul stated through Simon, was that she would be there when Jesus dies. Many maybe not. It certainly was not the only reason, but certainly may have been part of it. Lastly, I just thought the wise may have seen that star at the time of Jesus' birth, but after over a year and a half, they understood what that star meant to them, and that is when they identified it possibly. Conclusion: Each have their support of view, and I do not mind people believing it one way or the other, but for me, I found this way to be best. I am satisfied with it. I have showed you what I know and think. Wether you read it or not I do not know. However, these two possibilities are the only ones I found to be possible, in context. Lastly, pray that the Lord guides you through this, even if you have. He's been good to me. Praise God.

I read that Herod's palace was in Jerusalem. So this was where Archelus would rule from. My thinking is that after Jesus' dedication and Mary's cleansing at the temple, they went back home to Nazareth. They were either there when the wise man came, or visiting Elizabeth and Zechariah in Bethlehem, or rented or bought a house in Bethlehem for some reason. Then they fled to Egypt, came back after Herod died, but wouldn't want to stay in Bethlehem after what happened there. They might have thought Jerusalem safe, but being warned about Archelus, they went back to Nazareth. This tells me they spent some time in Egypt, perhaps they thought they had no home to go back to, initially.

The biggest flaw to your argument is "there could be a big time gap between Luke 2:38 & 39." Luke 2.27 & 2:39 both make reference to why Jesus was brought to the Temple:

27 And he came by the Spirit into the temple: and when the parents brought in the child Jesus, to do for him after the custom of the law,

39 And when they had performed all things according to the law of the Lord, they returned into Galilee, to their own city Nazareth

In both cases, reference is being made to "performing things after the custom of the law." This means that Jesus was being circumcised. Leviticus 12:3 makes the timing clear:

3 On the eighth day the boy is to be circumcised.

It is even clear that they brought the required gifts of a poor family for circumcision.

On the off-chance that the Wise Men got there within 8 days, they wouldn't have been poor. So, yes, it is safe to say that the Wise Men got there after Jesus' circumcision / Temple visit.

That said, typically, the tradition is that the Wise Men arrived before the Massacre of the Innocents - and since Herod had all those under 2 years killed, we arrive at "they got there within two years of Jesus' birth."

Birth of Christ Sequencing

From Nazareth to Bethlehem to Nazareth to Egypt to Nazareth

Mary espoused to Joseph, but he decides to put her away. God gives him a dream. After dream, Joseph and Mary are married.

Then Joseph being raised from sleep did as the angel of the Lord had bidden him, and took unto him his wife: Mat 1:24

We switch to Luke’s account. Joseph and Mary travel to Bethlehem as Matthew has given us the reason that they both go. Joseph goes per the decree of Caesar Augustus.

And Joseph also went up from Galilee, out of the city of Nazareth, into Judaea, unto the city of David, which is called Bethlehem; (because he was of the house and lineage of David:) Luke 2:4

Jesus is born.

And she brought forth her firstborn son, and wrapped him in swaddling clothes, and laid him in a manger; because there was no room for them in the inn. Luke 2:7

Angels inform the shepherds of the event.

And they [shepherds] came with haste, and found Mary, and Joseph, and the babe lying in a manger. Luke 2:16

8th day circumcision in Bethlehem

And when eight days were accomplished for the circumcising of the child, his name was called JESUS, which was so named of the angel before he was conceived in the womb. Luke 2:21

After the 33rd day of her required purification, they travel from Bethlehem to the temple in Jerusalem for offering.

And when the days of her purification according to the law of Moses were accomplished, they brought him to Jerusalem, to present him to the Lord; Luke 2:22

They meet two individuals Simeon and Anna.

And he [Simeon] came by the Spirit into the temple: and when the parents brought in the child Jesus, to do for him after the custom of the law, Luke 2:27

And she [Anna] coming in that instant gave thanks likewise unto the Lord, and spake of him to all them that looked for redemption in Jerusalem. Luke 2:38

After these things, the family returns to Joseph’s home in Nazareth.

And when they [Mary and Joseph] had performed all things according to the law of the Lord, they returned into Galilee, to their own city Nazareth. Luke 2:39

Sometime later, but less than two years after His birth, magi arrive in Jerusalem, searching for King.

Now when Jesus was born in Bethlehem of Judaea in the days of Herod the king, behold, there came wise men from the east to Jerusalem, Saying, Where is he that is born King of the Jews? for we have seen his star in the east, and are come to worship him. Mat 2:1-2

Herod wonders where the King is born and assumes wrongly that is also his residence. Herod calls his wise men.

And thou Bethlehem, in the land of Juda, art not the least among the princes of Juda: for out of thee shall come a Governor, that shall rule my people Israel. Mat 2:6

Herod now knows where the King was born and presumably still is, but next wonders when the King was born.

Then Herod, when he had privily called the wise men, enquired of them diligently what time the star appeared. Mat 2:7

Later we know Herod will have the male babies born less than 2 years of age killed in Bethlehem. So the star had appeared and they had travelled up to 2 years.

And he [Herod] sent them [magi] to Bethlehem, and said, Go and search diligently for the young child; and when ye have found him, bring me word again, that I may come and worship him also. Mat 2:8

This next verse is revealing and striking. The star had led the magi to Jerusalem where they thought the king would reign. They find out instead where he was born. To get from Jerusalem to Bethlehem, they just need ask for local directions. So, why then would the star appear again to them since again the magi now knew where Christ had been born? The only reason is because the family was no longer in Bethlehem, but in Nazareth as Luke informs. Don’t go to Bethlehem, but go to Nazareth. Thus the magi rejoice for the truth. Better to be led by God than by man.

When they had heard the king, they departed; and, lo, the star, which they saw in the east, went before them, till it came and stood over where the young child was. When they saw the star, they rejoiced with exceeding great joy. Mat 2:9-10

The contrast of time is also striking. The shepherds had found the family of Joseph, Mary, and Jesus the new-born the babe in a manger. The magi find the family in a house. They find a young child or an infant, not a new-born. Specifically, they find Mary and Jesus because presumably Joseph would be out working to support his new family.

And when they were come into the house, they saw the young child with Mary his mother, and fell down, and worshipped him: and when they had opened their treasures, they presented unto him gifts; gold, and frankincense, and myrrh. Mat 2:11

Rather than return to Jerusalem to tell Herod the news, the magi leave for their own country.

And being warned of God in a dream that they should not return to Herod, they departed into their own country another way. Mat 2:12

Joseph, though not present at their visit, is still the leader of the family. God warns him in a dream to take his family and leave Nazareth for Egypt.

When he [Joseph] arose, he took the young child and his mother by night, and departed into Egypt: Mat 2:14

Herod eventually realizes the magi are not returning to him.

Then Herod, when he saw that he was mocked of the wise men, was exceeding wroth, and sent forth, and slew all the children that were in Bethlehem, and in all the coasts thereof, from two years old and under, according to the time which he had diligently enquired of the wise men. Mat 2:16

After Herod’s death, the family leaves Egypt and returns to Israel.

But when Herod was dead, behold, an angel of the Lord appeareth in a dream to Joseph in Egypt, Saying, Arise, and take the young child and his mother, and go into the land of Israel: for they are dead which sought the young child's life. Mat 2:19-20

They leave Egypt and return to Israel and resettle in Nazareth.

And he [Joseph] came and dwelt in a city called Nazareth: that it might be fulfilled which was spoken by the prophets, He [Jesus] shall be called a Nazarene. Mat 2:23

This is the reconciliation of the times and events of those things surrounding Christ’s birth.

The wise men came after Jesus' dedication at the temple.

We know that after the wise men had visited, an angel appeared to Joseph and they fled immediately to Egypt and lived there.

Matthew 2:13-15 (NKJV) Now when they had departed, behold, an angel of the Lord appeared to Joseph in a dream, saying,“Arise, take the young Child and His mother, flee to Egypt, and stay there until I bring you word; for Herod will seek the young Child to destroy Him.”

When he arose, he took the young Child and His mother by night and departed for Egypt, and was there until the death of Herod, that it might be fulfilled which was spoken by the Lord through the prophet, saying, “Out of Egypt I called My Son. [Hosea 11:1]"

They did not return until after Herod's death. And though Joseph had wanted to return to Judea, they were warned by God to not to return there, and so they resided in Nazareth in Galilee, Joseph's hometown. Only then was the prophecy fulfilled that "He shall be called a Nazarene."

Matthew 2:22-23 But when he heard that Archelaus was reigning over Judea instead of his father Herod, he was afraid to go there. And being warned by God in a dream, he turned aside into the region of Galilee. And he came and dwelt in a city called Nazareth, that it might be fulfilled which was spoken by the prophets, “He shall be called a Nazarene.”

Luke 2:39 picks up from this period, after they have returned from Eygpt.

Luke2:39 So when they had performed all things according to the law of the Lord, they returned to Galilee, to their own city, Nazareth.

Luke 2:39-40 is a rapid summary of Jesus' childhood, while verse 38 relates to His dedication as a baby in the temple, Verse 39 tells of His return to Nazareth, and by Verse 41, it already jumps to His visit to Jerusalem as a youth at the age of twelve.

Here is my attempt to harmonize the accounts of Matthew 2 and Luke 2, This assumes a literal retelling of the story. (Luke as a careful writer has been vindicated again and again by archaeology.)

NAZARETH (above Samaria to the north) to BETHLEHEM

Luke 2:4 Joseph also went up from Galilee, out of the city of Nazareth, into Judea, to the city of David, which is called Bethlehem, because he was of the house and lineage of David,

BETHLEHEM (just south of Jerusalem)

Luke 2:7 Jesus born in a manger

Luke 2:8-20 Jesus still in manger when shepherds see them

Luke 2:21 Eight days for circumcision

JERUSALEM

Luke 2:22-24 Now when the days of her purification [33 days] according to the law of Moses were completed, they brought Him to Jerusalem to present Him to the Lord. [They gave gifts as poor people would, so the Magi had not given them riches at this time.]

Luke 2:25-35 Simeon sees Jesus.

Luke 2:36-38 Anna sees Jesus.

JERUSALEM to NAZARETH

Luke 2:39 So when they [Joseph and Mary] had performed all things according to the law of the Lord, they returned to Galilee, to their own city, Nazareth.

Approx. 1-year GAP. Then NAZARETH to JERUSALEM to HILL COUNTRY(?)

Luke 2:41 We learn from this passage that they went to JERUSALEM every year for the Passover. So we see them no longer in Nazareth, but in Jerusalem. Perhaps they turned aside to Bethlehem at that point to visit relatives and family. In Luke 1:39, Mary visits Elizabeth in the "hill country of Judah," which, according to this map, includes Bethlehem. After all, this is Joseph and Mary’s home town where their families could still be living, so it’s not inconceivable that they would go see them or stay with them during the Passover. Bethlehem is just below Jerusalem -- see the map above.

Matthew 2:1-8 “Now after Jesus was born in Bethlehem of Judea in the days of Herod the king, behold, wise men from the East came to Jerusalem,” [The Magi visit Jerusalem and Herod, where they are told by the scribes that the ruler is to come out of Bethlehem.]

Matthew 2:9-11 “When they [Magi] heard the king, they departed; and behold, the star which they had seen in the East went before them, till it came and stood over where the young Child was.” [This was not their home in Nazareth, but “the house” (Matt. 2:11), perhaps a family’s or relative’s home. This Magi’s visit was perhaps the first Passover after Luke’s account, for Herod said to kill children 2 years old and under.]

Matthew 2:12 The Magi return home without telling Herod.

BETHLEHEM to EGYPT

Matthew 2:13-15 “Now when they [Magi] had departed, behold, an angel of the Lord appeared to Joseph in a dream, saying, “Arise, take the young Child and His mother, flee to Egypt, and stay there until I bring you word; for Herod will seek the young Child to destroy Him.”

Matthew 2:16 babies killed in Bethlehem. [The slaughter may not have happened until several months after the Magi and family had left, when Herod finally wises up and causes those children 2 years old and under to be killed. It took time for the Magi to journey here from the east, so maybe they told Herod that the Ruler was born when they first saw the star? Adding that time and the time the Magi departed would come to about 2 years by Herod’s reckoning. If this is so, then the Magi may have come here at the first Passover after Luke’s account.]

Matthew 2:19 Herod dies. [The family survived in Egypt on the gifts of the Magi.]

EGYPT to NAZARETH

Matthew 2:22-23 Joseph is told to return, for Herod was dead. “…. And being warned by God in a dream, he turned aside into the region of Galilee. And he came and dwelt in a city called Nazareth, ….” Now we are back to Nazareth.