Mouthguards do not cure bruxism but are used to try and limit the irreversible damage done to your teeth from the grinding.
TMJ pain usually occurs on moving the jaw (opening or closing) and not necessarily on chewing. If you have issues with your TMJs, clicking, popping or cracking sounds may also occur during jaw movement.
If you are experiencing persistent pain only on chewing on the left hand side, I would suspect the possibility of a cracked tooth. This requires treatment at the level of the tooth and cannot be solved just by wearing a mouthguard.
I would also bring your mouthguard along for your check up to see if the plastic has become perforated. This is a sign that the mouthguard needs to be changed.
