The Byzantium Empire never recovered from the Sacking of Constantinople in 1204 by the Fourth Crusade, even when Constantinople fell back into Greek hands in 1261. From 1204 to long after 1453 when Constantinople fell to the Turks, you had Greeks stating they preferred Turkish rule to Latin rule. Osman and his successors use this hatred of the west to secure they hold on Anatolia. Many of the Ruling Elite Families of the Early Turkish empire were Greek Orthodox. Other Orthodox also supported the Ottoman expansion from 1299-1453 (The Greek Garrison of Constantinople was even occasionally hired by the Ottoman Empire as troops when the Ottoman emperors deemed the troops to be needed).
This is complicated by the Fact, Trapzon on the Black Sea Coast never accepted the Constantinople Rulers to be "True" Rulers of the Byzantium Empire and held out till 1461 against the Turks.
The last two Byzantium Emperors tired to get help from the West, but failed given that England was backing the Turks (More as a balance to France who backed the Greeks), as was most Italian States (Since 1204 the Italians had controlled the waters of Greece and feared Greek Expansion more than land based Turkish Expansion). Venice sent some troops, but not enough to stop the taking of Constantinople (and thus did NOT undo the Damages Venice had caused Constantinople when it directed the Forth Crusade against the City).
Given the hostilely of the Greeks to the West, do both to the Fourth Crusade and subsequent control of the Aegean Sea by Venice, many Greeks back the Turks. The Legend of Osman even mention this is that Osman first and best ally was a Greek Christan who helped him in his fight for control of what was to be the start of his Empire. Many of the land owners of Anatolia stayed Orthodox for generations after 1453, but over time came to think of themselves as "Turkish" not Greek (But he Duality of the Empire remained, with the Patriarch of Constantinople being appointed by the Sultan instead of the Emperor).
This duality remained the heart of the Empire till the late 1600s when a religious revival took place. This religious Revival was less religious in nature than an attempt to address the problem of the Empire no longer expanding. As long as the Empire was expanding, the elites had more and more land to divide among themselves, once the empire stooped expanding (about 1600) the elites had to divide up what already was in the Empire and with the Empire in Decline (late 1600s onward) and shrinking pot of land.
Like many societies in such a situation, people started to look for ways to keep a larger share of the shrinking pot. One way to gain an advantage is to make sure your sons get into a known power area. Thus in 1683 you see the switch in the nature of the Janissary Corp from one of sons of Christians (Soldier-Slaves) who as young boys were made soldier-slaves and who them moved up in the Turkish Bureaucracy to one of an organization of Sons of Muslims who worked the Bureaucracy. The Empire made other changes to Strengthened the Turkish/Islamic nature of the Empire. This was the result of the in-fighting of the Elites to gain an advantage int he declining Empire (A similar intolerance and demand for people of the "True Faith" occurred in Russia starting in the 1960s as the Communists demand greater and greater loyalty to Communist dogma as a way to force reformers and gain advantage for themselves within the Soviet Bureaucracy).
The problem with this is the Christian background of the older Janissary Corp kept the Turkish-Greek nature of the Empire alive. Once the the Janissary Corp became Islamic/Turkish from birth the Greeks lost power within the Bureaucracy. This lead to massive discrimination against the Greeks (and other non-Turks) within the Empire. Finally in 1830s the Greeks revolted and won their Independence (Egypt had been de-facto Independent since the French Invasion of the 1790s). The rest of Balkans followed as the 1800s went on. By the 1800s the "Turks" in these areas had ceased to be local representatives who happen to be Moslem, to be part of the repression of the people. In earlier Centuries, many of the locals had become "Turks" by inter-marrying with the Turkish elite and embracing Islam. By the 1800s these local elites had a choice embrace the Growing "Turkishness" of the Empire or embrace the locals being discriminated against. This caused these local leaders to either move to Constantinople to escape the problems back home, or to revert to whatever was the nationalism of they area they were ruling. Many families did both, but both caused the number of "Turks" o\in the balkans to drop in number. This is Similar to the Dropping numbers of "Russians" into today's Ukraine. The Ukraine is about 50-50 Ukrainian-Russian in nationality (Under Soviet Rules you had to declare your nationality when you turned 21, most mixed parentages children under Soviet Rule picked Russian, but are in outlook Ukrainian, the same with the "Turks" in the Balkans in the 1700s and 1800s, many were NOT Turks in the Modern sense of the World, but picked the nationality of Turk as an political advantage. When it was no longer an advantage the term was dropped.
For more on Osman I see:
http://en.wikipedia.org/wiki/Osman_IThe Forth Crusade:
http://en.wikipedia.org/wiki/Fourth_CrusadeTrabizon:
http://en.wikipedia.org/wiki/Trabzonhttp://www.karalahana.com/english/archive/history.htmlJanissary Corp:
http://en.wikipedia.org/wiki/Janissaryhttp://en.wikipedia.org/wiki/Devshirmehhttp://en.wikipedia.org/wiki/Janissary_revolts