Reading Book Three, it seems that time turners cannot change the past. I never finished reading A Cursed Child to see if it explains the contradiction, but that there is a contradiction to explain is one of the many reasons why I refuse to consider that as much more than a fan fiction.
The true fan fiction work Living Amongst Ghosts tries to provide a way to let someone return from the past without breaking Book Three’s rules: break the time turner while in the past, and the magic returning you to the past is broken, ejecting you from the (past) time stream and returning you to your “proper” place in time.