Starter not turning engine can be two things (assuming battery is good).

1. Bad earth connection. The starter gets its connection to the negative battery terminal via the chassis. Try running a thick wire or earth strap from the battery negative terminal straight to the mounting bolt on the starter.

2. Worn commutators inside the starter. While the starter may seem to run fine with no load against it, it may have worn electrical contacts inside that can't pass enough current to turn it under load.

Fixing # 1 is easy and cheap. Fixing # 2 means a starter rebuild by an auto sparky.

I've bought a starter from a wrecker which he tested on a battery only to find it wouldn't start my car and had to do the rebuild.

Good luck!