Your starter or the leads to it may be bad. If the engine will turn by hand with the plug in it should also turn with the starter.

Does the compression seem right when turning by hand?

Someone has already asked about the washer under the plug (if missing it could hit the top of the piston).

If I read you correctly you said it would turn a few times with the starter and then stop, that sounds as if it is getting flooded.

I have run out of ideas.