I'm not sure. I know what the previous poster was getting at, but you're asking because you have had a diagnosed thyroid condition that you're getting treated for.
Are you feeling alright otherwise with the dosage you're taking? Perhaps you should first see where your thyroid levels are at and take into consideration how you feel overall.
I know why you ask, because I've had to deal with this issue since hitting puberty. If it were only a few hairs it wouldn't seem so bad. Shaving is a nuisance with curly hair and sensitive skin, too, because of the tendency for the hair to become ingrown. And, as remarkable as the idea of laser treatment sounds, like a permanent solution, it's not affordable for a lot of us. So, out come the tweezers and other old fashioned methods of hair removal.
http://www.sharecare.com/question/causes-facial-hair-women