My oldest started at 6 and did ok mentally at first, then would break down before the match even started so we decided to not go to any more tournaments his first year (he went to maybe 3 or 4 his first year). We continued going to practice, and didn't attend a single tournament his second year...just practice. His 3rd year we only went to a tournament every other week. Now he is fine. I had to handle him with kid gloves.
My youngest started last year, and like what was mentioned above, he has been going to tournaments and watching kids wrestle for a few years, and he hasn't had any problems (other than keeping his head on the mat and not moving fast enough,...lol)
In my opinion, roll with the punches and take what you can get. If your child seems overwhelmed, don't let him quit. Instead, just go to all the practices and have fun. He will eventually grow out of it.