Why didn't "Europe" support the Byzantine Empire against the Turks?

Actually, "Europe", understood as a mix of nations that were Christian and occupied the european continent, did support the Byzantine Empire against its threats, especially Muslim, and especially Turkish threats.

Against Muslim forces from the differents califates, the Byzantine Empire beneficiated from time to time of the help of Italian republic forces.

When the Seldjukids Turks threatened the Byantine Empire, Bulgarian and Kievians forces helped the Byzantine Empire to ultimately repell them, before the Seldjukids were destroyed by colliding forces.

Against the Ottomans, Constantinople was helped by an European coalition, that fought and was destroyed in the battle of Nicopolis. THis was a huge effort: France lost so many knights during this battle that it was weakened and suffered from a reboost of the Hundred Years War.

The question needs a time period, but assuming that you are asking about the period when Constantinople fell in 1453, the old map below shows the situation:

As you can see from the map, Constantinople was in a desparate situation, completely surrounded by Turkish territory for hundreds of miles. The only European power that could help them was the Republic of Venice, which still controlled the sea route to Constantinople, a slender thread of a lifeline. Venice controlled the legendary port of Negroponte, shown as a red dot on the map, and by this link connected to Constantinople and beyond to Trebizond in the Black Sea. This valuable commercial route was kept open despite being completely surrounded by Turkish power. Venice did what it could, but eventually the Turkish forces became overwhelming.

In 1054, The Great Schism occurred; this was an official "split" or in reality, a mutually agreed divorce between The Roman Catholic Pope-(i.e. "The Bishop of Rome") and the Greek Patriarch of Constantinople. More specifically, The Great Schism was a split or divorce between The Western and Eastern rite Churches.

Although the Byzantine Empire was a mixed Roman and Greco-Roman Empire during the first 275 years of its imperial reign, it became an increasingly Greek Eastern Christian Empire beginning in the early 600's with the crowning of Emperor Heraclius. From 606, until 1206, the Greco-Byzantine Empire dominated the Eastern Christian world and grew in power and influence. The city of Constantinople was one of the largest cities in the world and Constantinople was, by far, the largest and most sophisticated Christian city in the world at that time. For the Pope, the Papal states, as well as the (fairly long) line of Holy Roman Emperors, Constantinople, was the prize package for an invasion. The Byzantine citadel, would be an important preliminary conquest prior to reaching Muslim Palestine-(specifically, Jerusalem). The Crusading Armies from the West-(with blessings from the Pope) conquered Constantinople in 1206 and occupied the Byzantine Citadel until 1256; this marked the historic turning point for Medieval Greek Christian Constantinople.

Beginning in 1256, the newly liberated city of Constantinople now had to face the Seljuk Turkish Muslim threat moving closer to the city, as well as towards Western Asia Minor. Although Constantinople regained its independence from the Crusaders, it was unable to return to its preeminent status and in turn, the Power and Governance of the Byzantine Empire relocated southward to the Peloponnese, specifically to the interior town of Mystras-(near ancient Sparta). From 1256, until, May, 1453, Constantinople-(similar to Rome after the Pax Romana 1000 years earlier), became a figurehead city, as well as a militarized city on the constant defense from Seljuk Turkish Muslim encroachment and ensnarement. Though Constantinople, despite its beleaguered status, continued to remain the Capital of Eastern rite Christianity. However, by May, 1453, it was curtains for Constantinople and the city was overrun, conquered and occupied by the Seljuk Muslim Turks.

Although there appears to be a spirit of religious ecumenism and reconciliation between the Eastern and Western Churches since the historic 1965 Second Vatican Council, one must not forget the deeply rooted bitterness and enmity that existed between the 2 Churches since 1054 and especially since 1206. The Papacy hated Constantinople and the feelings of the Byzantine Leadership towards the Papacy in Rome were mutual-(This MAY have been the time where the word, "Byzantine" became a part of the Western Dictionary, though pejorative in meaning when read by Western eyes).

By the 1400's, Greek academic, financial and even a few religious elites escaped the Seljuk Muslim Turkish advancement towards Constantinople and relocated to the Veneto in Northeast Italy, specifically, Venice. Conversely, the Venetians, had occupied a sizable portion of Greece's islands and the Peloponnese-(even during the early years of Ottoman occupied Greece). Although the Greek newcomers to the Veneto had several precious ancient manuscripts to offer the burgeoning Northern Italian Renaissance Scholars and Luminaries, living as a Greek Eastern Christian in an increasingly Western Roman Catholic world was overwhelming and obscuring. In other words, Roman Catholicism, as a theology, a political presence, as well as a major cultural Mover, had triumphed over Byzantine Christianity with The Fall of Constantinople in May, 1453-(However, such a triumph for the Catholic Church proved to be somewhat ephemeral in the wake of a rising Russian Orthodox Christian global presence, as well as with the rise of Martin Luther and the Protestant Reformation).

The elimination of the Byzantine Patriarch's global power, as well as the elimination of the Byzantine Emperor's power, led to the Papacy's consolidation of theological and territorial power West of Constantinople......but NOT to the (immediate and semi-distant) NORTH of Constantinople.....these territories became part of the larger Ottoman Turkish Muslim Empire.

So there are many reasons as to why the European West remained distant from the Muslim-Christian struggle for power in Constantinople during the Late Middle Ages and Early Modern era. There is no singularly rooted answer to this question; it is complex in its explanation and continues to engender great debate to this day.

Please keep in mind that the IVth Crusade mentioned in the first answer has resulted in taking of Constantinople by mostly Venician troops in 1204. This has resulted in a long-lasting civil war between the Latins and the Byzantines. Finally Constantinople was taken back by the Byzantines in 1261, but the Empire did not regain all its territory and wealth.

Facing the Tukish menace, Byzance has tried to gain the help of Rome by uniting with the Latin Church. But, though the Orthodox Churches have signed the "Laetentur coeli" union with the Latin during Ferrafa-Florence Council in 1439, this union was often rejected by the clergy and the population. In fact, the Union weakened the Orthodox Church. The Greek Church has repudiated its signature. In 1441, the Russian Church proclaimed its independence, imprisoning the Kiev Patriarch (who has after become a cardinal and later, the Latin Patriarch of Constantinople). In 1451, even the unionist Bysantine Patriarch has to seek refuge in Rome. The subsistence of the "byzantine schism" is the formal pretext for the Pope Nicolas V not to help Byzance. After the Union was proclaimed in Constantinople in december 1452, the Pope has changed his mind, but his call for a new Crusade met no success.

The latin world was discouraged by the defeat in Varna in 1444. By 1453, Hungary and Venise were bound by numerous treaties with Mehmed II the Conqueror. The French and the English were in war agaist each other. Finally, only Venice and Gene have sent some limited help.

There was a certain amount of natural antagonism between the west and the Byzantines. Part of this was religious: They belonged to different sects of Christianity, and thus often viewed each other as little better than heretics or Muslims. Another part was commercial. What little commerce the west had was in direct competition with the Byzantines, whose geographic position forced Western merchants to pay whatever fees were demanded (or try to sneak through the Bosphorous).

Even so, in March of 1095 relations got warm enough and the Byzantines got desperate enough to ask the west for help. This was just a simple request for mercenaries to help against the Seljuks.

However, Pope Urban II had a more grandiose vision in mind. He convened The Council of Clermont, where he argued that it was a Christian moral duty to aid the Byzantines against the Muslims. He backed this up with a edict forbidding Christians waging war on Christians except on alternating weekdays(!), and with a rousing speech that was distributed and publicly read by the clergy throughout the Catholic area.

The result was the Crusades, which lasted for roughly 2 centuries, and one of which ironically ended up administering the coup-de-grace on the Byzantine empire.