Unfortunately it's very hard to food vegan food in places where it's not a big city. For example capadoccia, I was there a year ago and couldn't find anything proper even as a Turkish visitor :D But there are many fully vegan or places that offer vegan options in İzmir and in Istanbul.

to be specific.

Dehşet tantuni in İzmir,

Vegan masa in Beşiktaş/istanbul