Doing the google search you suggested, I didn't find anything suggesting that carnitine CAUSES hypothyroidism. It seems that carnitine MAY act as an agonist to T3 and T4. If you are hyperthyroidism, this may improve the situation. If you are hypothyroidism, agonizing T3 and T4 may not be a great idea. In either case it doesn't change the levels of t3/T4 so tsh and the negative feedback loop shouldn't be effected is my guess. Of course, I'm uneducated and could be wrong, but nothing I see suggests carnitine as a cause.
I also saw some information that showed carnitine helped normalize function of both hyper and hypothyroidism subjects.