My limited understanding is: The onset of starting thyroid medications for low thyroid levels may initially reduce any (suboptimal) body production of thyroid hormones that is still naturally occurring. So until the pill form of thyroid medication you have started gets optimized in your system (a often slow process) improvement isn't forthcoming until then. You may actually feel worse before feeling better.

Dedicated thyroid online groups I have read will tell you that thyroid hormones can not work well unless your body's vitamin D, folate, ferritin and vitamin B-12 levels are in optimal ranges. They recommend testing them once a year (even if you end up having to pay for self testing). Should you have to self test, it is best to have a doctor review results nonetheless before supplementing so that any underlying condition does not go undiagnosed. If say, one's vitamin level is low, prescriptions to cover the cost of a key vitamin such as these should be offered.