I just started a new world and spawned on an island, and the Turtles that spawn next to me and on the islands nearby stay stuck on one block next to the Water. 

Turtles will stay on the block and stay there until you push them or kill them. They will still look around, make noises and breed when you give them Seagrass, but other than that they stay stuck in one block and don't move an inch.