The last time inquired about mooring a small boat i was quoted roughly 500tl ayear at the mariner by the BP garage. As long as the boat is less than 5mtr long and the outboard is under 10hp you do not need any licence or need to register it.There is some smaller places around the other side of Fethiye where you can speak to the Restaurants that have jetties and work out deals regarding mooring there. There is also the small mariner by the last restaurant on the prom which i think is free it is worth asking at many places around the seafront but it depends on the location you feel is best for your needs.
As Wordbird says, no matter how small your boat is, from personal experience, you WILL need to have a licence to drive it , IF you wish to stay within Turkish Maritime Law. Please beleive me, The Coastguard do not, and will not mess around !! Also, I would think very, very carefully before taking up a "cheap" mooring offer at ANY of the numerous restaurants or bars on the other side of Fethiye Bay, I used to keep our boat in our driveway, until I was offered " free ", yes, FREE, mooring in the little fresh water river which empty's into the sea at Karatas. The river mooring was the clincher, this obviously meant that I could flush the motor through with fresh water after each use, therefore preventing the motor from "salting up "!! I even got the offer of free fuel , from large drums at the rear of the cafe !! All was well until one day Lorraine and I drove around to take the boat out, and it was gone ! After 15 minutes, we saw OUR boat being used by Turkish fishermen in the bay. We had one hell of a job convincing these guys that the boat did, indeed belong to us. The next time we went to use the boat, there was a bloody great hole in the bottom ( double skinned cathedral hull ), the cafe owner said ; " big sea, he came, smash boat ". I took the boat back to our driveway, and subsequently sold it !! BE WARNED !!