1- Turn your face right and then left.
2- Turn your face to the right and then to the left.
'turn' here is being used as a phrasal verb, which is 'turn (something) to the..'
A doctor examining something about a person's face might say, for brevity, (1), but it would be understood that it is just a quick short form of (2), so use (2) only.
3- Turn your face to the right side and then to the left side.
To the right side of what? The sentence is asking the person to turn his face in relation to the right side of some specific external object. Just how one would actually do that I can't imagine. How do I "turn my face to the right side of the dog/door/school/John' ?
You would have to say, turn your face to its right side, but this is clumsy, awful, and unnecessary when one can just say (2) above.