When taking thyroid medication, TSH is a useless test. There are several reasons why TSH has only a weak correlation with the actual thyroid hormones, and a negligible correlation with hypothyroid symptoms, which should be the real source of patient concerns.
When taking thyroid med, it spikes thyroid levels for the short term and thus has a suppressive effect on TSH for most of the day. The drop in TSH reduces output of natural thyroid from the gland. The net effect is basically no change in thyroid levels until TSH is essentially suppressed. At that point further increases in med dosage will start to raise thyroid levels. This necessitates a full daily replacement amount of thyroid hormone in order to feel normal.
Your thyroid status is mainly associated with levels of the biologically active thyroid hormone, Free T3 (FT3), which regulates cellular metabolism. The other main thyroid hormone Free T4 (FT4) is a prohormone available for conversion to T3. Note that hypo patients taking only T4 med often find their body does not adequately convert T4 to T3. A good thyroid doctor will test and adjust FT4 and FT3 levels as needed to relieve hypothyroid symptoms, without being concerned about TSH levels. Symptom relief is all important.
In addition most hypothyroid patients find they are deficient in Vitamin D, B12 and ferritin. all are important for best results. D should be at least 50 ng/ml, B12 in the upper part of its range, and ferritin at least 100.
If you want to confirm all this, have a look at my paper in the following link. If you have any difficulty in getting tested for FT4 and FT3, and treated for symptoms, you might try giving a copy to your doctor.