TSH is a pituitary hormone that is affected by so many things that , except at extreme levels, it is inadequate as the sole diagnostic for thyroid status. You should always be tested for both of the biologically active thyroid hormones, Free T4 and Free T3, which are not the same as Total T4 and Total T3. Your TSH levels are high enough to indicate the possibility of having Hashimoto's Thyroiditis. with Hashi's the autoimmune system erroneously identifies the thyroid gland as foreign to the body and produces antibodies to attack and eventually destroy the gland. As this is occurring, the output of the gland is diminished and in response the pituitary produces increasing amounts of TSH.
If not tested beyond TSH, you really need to be tested for Free T4, Free T3, cortisol, Vitamin D, B12 and ferritin. In addition you need to be tested for the antibodies associated with Hashi's. Two tests are required: Thyroid Peroxidase antibodies (TPO ab) and Thyroglobulin antibodies (TG ab).
The small dose of T4 thyroid med will not make a significant difference for you. The reason is that the med will cause a reduction in your TSH output which will reduce output of natural thyroid hormone. Your serum thyroid levels are the sum of both natural thyroid and thyroid med. Only when your med dosage is high enough that TSH is suppressed enough to no longer stimulate natural thyroid hormone production, will serum thyroid levels reflect further increases in thyroid medication. In addition you may find that your body does not adequately convert T4 to T3, resulting in inadequate Free T3 levels.
A change in diet is not going to fix this for you. You will need to get the additional tests done and also get the doctor to increase your med dosage every 5-6 weeks to get your Free T4 to at least mid-range, and probably add some T3 to your med to get your Free T3 into the upper third of its range. In addition you need to test and supplement Vitamin D, B12 and ferritin as needed to optimize. D should be at least 50 ng/mL, B12 in the upper end of its range, and ferritin should be at least 100. You can confirm all this by reading at least the first two pages of the following link, and read more if you want to get into the discussion and scientific evidence for all that is recommended.
https://tinyurl.com/y8k2zswb
The single most important thing for you is to find a good thyroid doctor that will test and adjust Free T4 and Free T3 as needed to relieve symptoms, without being influenced by resultant TSH levels. Symptom relief should be all important, not just test results. Note that a good thyroid doctor does not necessarily mean an Endo.