Thyroid medication is rarely the entire solution for anyone because of how badly thyroid conditions affect absorption and metabolism of nutrients.
Stomach acid is low in hypothyroidism. Some people get no relief from taking thyroid medication simply because they are not even able to absorb the medication. You need stomach acid to do this. Usually they tell you to take meds on an empty stomach and not to eat anything for a few hours. But if you do that, you don't trigger the release of any stomach acid.
It's best to take thyroid meds with something acidic, like vinegar (yuck though), or lemon juice, or just citric acid (vitamin C).
My personal journey in healing my thyroid did not involve taking thyroid meds. I recovered with Iodine and MANY other vitamins and supplements. I'm proof it is possible. All the B vitamins were really important in my recovery. Iodine was only one small piece of the puzzle. Looking back, I may have recovered faster if I'd taken thyroid medication in addition to all the vitamins and supplements. There are many paths to recovery, but always always take every vitamin, not just a few, and take extra B1 (thiamine), possibly the most universally important vitamin for thyroid folks, and do what you have to do to increase your stomach acid.
Too much time with low stomach acid can lead to bacterial overgrowth (SIBO), which can contribute to the vitamin deficiencies you already are struggling with because of the thyroid issue.
Low stomach acid means not breaking down food. Bacteria takes over in the small intestine where we don't want it to be, and steals vitamins. We become deficient in lots of things, and usually in the B-vitamins first. B6 gets low. B6 is needed to get Iodine into the thyroid. So you become iodine deficient even if you have iodine in your diet. B1 (thiamine) gets low, and you can't make stomach acid without it. And the cycle repeats.
Start with thiamine. If you get any rise in your thyroid from anything else you take, you can quickly become depleted in thiamine, which can be very dangerous. Search for BeriBeri (thiamine deficiency), and you may recognize a lot of thyroid symptoms in the description.