Try the guys halfway along the road where all the seafood restaurants are in Yalikavak. They made all our furniture for us and were wonderful, real craftsmen. It is, of course, slightly beneath thier carpentry skills to build a mere shed but I doubt they would turn away work.
One tip though, always take a clear sketch or photo of what you want along with measurements written onto the actual picture - makes it clear for everyone.
Also be aware that timber does seem expensive in Turkey, can't quite figure out why when you see the miles of pine forest but there you go!