4.3 is a little high. I start losing hair with that TSH. Did they check your T3 and T4 levels?
I would say check iron also, but on your CBC they should be able to deduce if your iron is low even of they did not directly test iron. If they did blood tests I can’t imagine they did not run a CBC, but you never know. Are your lips, tongue, and inside the lids of your eyes very pale?
I would see an endocrinologist. I would get more thyroid testing and let an endocrinologist interpret the tests, not a GP or an internist. Many labs still have up to 5.5 as normal for TSH, and that is very high. There have been arguments to change the parameters within the medical community, and endocrinologists would never let a patient hover that high, they would change the medication dose. Your thyroid could be bouncing around, so frequent testing might be worth doing over the next few months. Test at least once a month to see if your numbers are outside of “normal” at times.
It might be something else besides your thyroid, but 4.3 is a red flag in my opinion.
People swear by biotin for hair and nails, you could try taking some while you also see another doctor.