“Sur la gauche/droite” refers to an area which is on the left/right side. “À gauche/droite” refers to what you can reach by going in a left/right direction. The two are synonymous when talking about the position or something, but sur doesn't really work when talking about the direction something is headed towards.
“La bibliothèque se trouve sur la gauche” and “la bibliothèque se trouve à gauche” are mostly synonymous. There is a preference for sur in some cases, for example “la bibliothèque se trouve sur la gauche sur cette photo” means that the library is in the left 45% or so of the picture, whereas “… à gauche …” would exclude more of the middle. On the other hand, à is somewhat preferred in sentences like “la bibliothèque est à gauche de l'église”, but “la bibliothèque est sur la gauche de l'église” also works; both allow the library to be either adjacent to the church or further to the left.
“?Tourner sur la gauche” is not really idiomatic because turning is about the direction you're facing, not about your position. I would never use this when giving directions, for example. This sentence makes me think of the way you're turning, like which leg you're using to pivot in a dance move. I think there's some variation between native speakers there, even within France, though. Checking online, it seems there are people who use “tourner sur …” in a car. But it's not something I would say. Of course it's perfectly comprehensible.
