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.
Well, now you’re adding “on life”, which you didn’t mention before. Just look it up, OK?