A small idea, how time turners may work, or how it can be described in the future chapters:

Let's assume that real travel back in time is not possible. But we can have mysterious source of magic somewhere, and this thing can be supercomputer or (and) superintelligence capable of correct prediction of events that will happen in some limited volume of space and time. And this superbeing can predict activation of time turner in future, predict mental and physical state of the person who will do that and create a copy of this person in the present time. "Magic" happens.

P.S. Wild guess: source of magic or one of its parts is somewhere inside Azkaban rock, just because time turners does not work there, which may be self defence mechanism.