If the rash appeared before you started synthroid but has since subsided, it might be responding positively to the meds. (I.e., your skin might have been more sensitive because it lacked T3 and now the med is taking care of that.)
An interesting anecdote: my husband was having a terrible time with a rash that appeared under his arm, and he was afraid it might be tinea cruris spreading (he'd been struggling with the latter for years). I persuaded him to talk to his doc about trying low-dose Armour thyroid (for other reasons) and --voila!-- not only did he feel better than he has in decades, but his skin completely cleared up!
Not many docs know much about thyroid/skin connections, but it's definitely something someone should do research on!