Just to be interesting, I'm going to suggest the Byzantines manage to expel the Turks entirely from Anatolia sometime in the reign of John II or Manuel I, but the Empire remains fairly shaky once Manuel dies. In 1207, when the Third Crusade arrives (let's say the stronger Byzantium delays the Muslim reconquest of Jerusalem to circa 1200), the Empire is unable to resist the Crusaders and collapses, to be partitioned into Crusader States.
By the year 1800, when I'm guessing the "modern era" kicks in, we have an Empire of independent states under the nominal suzeiranty of the Roman Emperor in Constantinople, who rules the Balkans and Western Anatolia directly, exerting more or less feudal pressure on his Eastern vassals as time goes by. These are as follows.
Despotate of Trebizond (Orthodox)
Despotate of Sebastea (Orthodox)
Armenian Principality of Cilicia (Monophysite)
Duchy of Iconium (Catholic)
Duchy of Caesarea (Catholic)
Despotate of Pontus (Orthodox)
Turkish Kingdom of Attaleia (Orthodox)
Kingdom of Antioch (Catholic)
Kingdom of Jerusalem (Catholic)
Well, at least there's something a bit different for you all to mull over...