You could try
http://www.thk.org.tr/2005/anasayfa.htm Turk Hava Kurumu, THK, the Turkish Air Club. There are also two private companies, Onur air and Pegasus, but I don't think they rent. THK does, and they'll work with you. Call their main office in Ankara. They might ask you to go to their Efes-Selcuk airport (LTFB) near Izmir since they train pilots there. It won't be cheap though. According to the rules, there must be a Turkish national in the cockpit.
If you see an old 172 with the registration TC-MAP on the ramp, that was my bird. I sold her to Bodrum Imsik Airlines in 96, but I think she was sold onward some years later. I was the first foriegn owner of a private aircraft in Turkiye. It's a beautiful coast to fly over though. Enjoy.