How does Jesus show love by suffering?

I wonder, how does the suffering of Jesus/God show love.
I cannot believe that Jesus suffered to forgive our sins.

Indeed, his love should be defined by more than whether Jesus died to forgive our sins. Rhoads, Dewey and Michie say, in Mark as Story, page 113, that the modern reader needs to be cautious not to read into Mark theological meanings that later came to be associated with Jesus' death. Although their admonition referred specifically to Mark's Gospel, the same of course applies to the other gospels of the New Testament. And this is where the authors address this part of our question (ibid):

Mark does not portray Jesus' death as a sacrifice for sin. Mark portrays Jesus already pardoning sin during his life and authorizing others to do the same. His death is not needed to make forgiveness possible.

Acts 10:43 assures us that belief in Jesus is the sole prerequisite for forgiveness of sins, with no suggestion here that his death was a prior necessity: "To him give all the prophets witness, that through his name whosoever believeth in him shall receive remission of sins."

A single New Testament reference explains the belief in forgiveness of sins through the death of Jesus (Colossians 1:14): "In whom we have redemption through his blood, even the forgiveness of sins." In Who Wrote the New Testament, page 183, Burton L. Mack says the letter to the Colossians is not authentic, with most New Testament scholars dating the epistle to the 70s of the first century, which consequently places some doubt on the authenticity of the epistle's theology.

Bible Life Ministries states that the Bible says that on the cross Jesus took upon Himself the sins of the whole world, which is of course a reference to Colossians 1:14. Colossians links salvation (redemption) to the death of Jesus, but Bible Life Ministries says that is false, since individual salvation comes from faith in the resurrection of Jesus. Charles Swindoll (How Does the Death of Jesus Save Me) says that we are saved because Christ died on the cross as our substitute, and calls this 'substitutionary atonement'.

To have died willingly for the forgiveness of sins must be an act of love, if our sins could never otherwise be forgiven. The belief that the crucifixion was necessary for the forgiveness of sins seems to have started with Colossians, an epistle of uncertain origin. As stated in the question, this is really inconceivable.

How loving is it to not give someone justice for being wronged? If I killed your close family member, would it be loving for someone to simply say "that's too bad" and not atone for the murder of your family?

Love does not just require grace, but also justice. Doing wrong has a very real cost, both emotionally and physically and that slack has to be picked up somewhere. By taking the suffering and wages (cost) of sin on Himself, Jesus resolved the wrongs done to others with both grace and justice.

Justice in that the costs were paid but grace in that he choose to fix it rather than requiring us to.

Note: The quotes I offer here are taken from the 1983 Catechism of the Catholic Church. There is an online version here. Numbers are the paragraph numbers for each point.

601 The Scriptures had foretold this divine plan of salvation through the putting to death of “the righteous one, my Servant” as a mystery of universal redemption, that is, as the ransom that would free men from the slavery of sin. Citing a confession of faith that he himself had "received," St. Paul professes that "Christ died for our sins in accordance with the scriptures." In particular Jesus' redemptive death fulfills Isaiah's prophecy of the suffering Servant. Indeed Jesus himself explained the meaning of his life and death in the light of God’s suffering Servant. After his Resurrection he gave this interpretation of the Scriptures to the disciples at Emmaus, and then to the apostles.
602 Consequently, St. Peter can formulate the apostolic faith in the divine plan of salvation in this way: "You were ransomed from the futile ways inherited from your fathers... with the precious blood of Christ, like that of a lamb without blemish or spot. He was destined before the foundation of the world but was made manifest at the end of the times for your sake." Man's sins, following on original sin, are punishable by death. By sending his own Son in the form of a slave, in the form of a fallen humanity, on account of sin, God "made him to be sin who knew no sin, so that in him we might become the righteousness of God."
604 By giving up his own Son for our sins, God manifests that his plan for us is one of benevolent love, prior to any merit on our part: "In this is love, not that we loved God but that he loved us and sent his Son to be the expiation for our sins." God “shows his love for us in that while we were yet sinners Christ died for us."
605 At the end of the parable of the lost sheep Jesus recalled that God’s love excludes no one: "So it is not the will of your Father who is in heaven that one of these little ones should perish." He affirms that he came "to give his life as a ransom for many"; this last term is not restrictive, but contrasts the whole of humanity with the unique person of the redeemer who hands himself over to save us. The Church, following the apostles, teaches that Christ died for all men without exception: "There is not, never has been, and never will be a single human being for whom Christ did not suffer."

This gets into the whole "original sin" discussion, which is a bookful in itself. But I hope the idea is clear: There is a sort of debt we owe as humans, something that each of us owes individually and collectively. What God did for us is to show his love for us in the deepest and most intimate way possible, by becoming one of us, and as one of us giving up his life to clear this debt.

One more quote from the Catechism:

618 The cross is the unique sacrifice of Christ, the "one mediator between God and men." But because in his incarnate divine person he has in some way united himself to every man, "the possibility of being made partners, in a way known to God, in the paschal mystery" is offered to all men. He calls his disciples to "take up [their] cross and follow [him]," for "Christ also suffered for [us], leaving [us] an example so that [we] should follow in his steps." In fact Jesus desires to associate with his redeeming sacrifice those who were to be its first beneficiaries. This is achieved supremely in the case of his mother, who was associated more intimately than any other person in the mystery of his redemptive suffering.
Apart from the cross there is no other ladder by which we may get to heaven.

By Jesus' sacrifice, then, we're invited to accept the possibility of sacrifice in our own lives. Not just any sacrifice, either. I mean, if God Himself could sacrifice His Own life, then how big, really, are the little sacrifices we are asked to do? Even sacrificing our own lives is not too big for God to ask.

There's a lot more to Jesus' death for us than this, and a lot more (for example) in the Catechism, which you might want to take a quick look over. Here is the beginning of the section (the "article") on the death and resurrection of Jesus.

It looks like I did more actual quotation than explication here; but I hope this serves as a beginning at least.