you can`t...

and the best way to go is to forget about this issue.
everything that is shown over the internet can be copied at the sole discretion of ones will.
even if you had a great idea, and even if you watermark it, and post it at a low resolution, or very small or whatever, another less talented and less creative designer that does know how to handle the tools CAN remake it.

so the best solution for your peace of mind is to get used to it. besides, if it gets stolen it means it is worth it, liked and you did a great job ;)

the fun part is when you discover where your stolen work appeared. For those situations is good to always keep your source files and maybe even the right screenshots on where it was first posted.

good luck

I've entered as well as hosted many logo/design competitions.

Copyrighting your design is a waste of time. If your logo is chosen as the winner, you will have to revoke your copyright - it's probably written into the rules ("All submissions become the property of __") By entering the contest, you accept these terms. The design will belong to the contest holder just as if they had hired you to design it, so I wouldn't even bother.

Quite often the final design is reworked anyway, by you or another designer. Regardless, the only thing you're legally entitled to receive is recognition as the contest winner, and the prize.

Watermarking a submission for a competition is not a great idea. Not only does it appear amateurish, the judges will want the image to be as clear and understandable as possible.

A low-res file is the very best idea. Good luck!