The Turks were dominant in much of what is now eastern and central Turkey after their victory in 1071 at Manzikert against the Byzantines (for a while the Byzantines held on the the westernmost part of Anatolia). By the early 1400s there wasn't much left of the Byzantine Empire except for the area around Constantinople and parts of Greece--a number of the Greek islands were under Western Christian rule (mostly Venetians). Cyprus and Crete were under Venice for a long time and free of Turks (other than attackers--there was a raid on Crete in 1571 not long before the famous battle of Lepanto). Venice held on to Crete until 1669.
I don't think there were any Turks at all on Cyprus before 1570, other than those making attacks, not living there.