The expression “turn over a new leaf” is the correct one. It comes from when a page of a book was called a leaf, so you can see how turning to a new page can be a metaphor for making a change. “Turn over a new lease” doesn’t really make much sense, certainly not as much as “turn over a new leaf”.
You’re right that expressions get changed according to what people think they hear, especially when people haven’t seen the expression in print, or don’t think about what the expression means, or both.
When you put the saying with ‘on life’ it does. “Turn over a new lease on life” (which is how it is normally used) definitely does make sense, far more so then “Turn over a new leaf on life” which doesn’t.
It means you’re signing a ‘new lease on life’, meaning you’re changing your ways and starting anew.