Before further discussion, we need to know if you have been diagnosed hypothyroid previously and are taking thyroid med?
Your doctor is making the usual mistake of deciding you are over-medicated and hyperthyroid just because your TSH is suppressed below range. This is totally wrong because in the untreated state our bodies have a continuous low flow of thyroid hormone from the gland. Each individual has an equilibrium among TSH, Free T4 and Free T3. When you take a significant dose of thyroid med once or twice a day, it affects the equilibrium and TSH becomes suppressed for an extended period. So, when taking thyroid medication, a suppressed TSH does not mean you are hyperthyroid unless you have hyper symptoms due to excessive levels of Free T4 and Free T3, which is not your case. There are even scientific studies that show TSH usually becomes suppressed when taking thyroid medication adequate to relieve hypo symptoms. If you want to confirm what I say click on my name and then scroll down to my Journal and read at least the one page Overview of a paper on Diagnosis and Treatment of Hypothyroidism: A Patient's Perspective.
So you need to get your doctor to understand this and continue to increase your thyroid med dosages as needed to relieve your hypo symptoms. If you expect difficulty in getting this done, then give the doctor a copy of the paper and ask him to read and reconsider. If you do not succeed with that, then you will need to find a good thyroid doctor that will do so.
Also a couple of other things. Hypo thyroid patients are frequently deficient in Vitamin D, B12 and ferritin, so you need to get those tested and then supplement 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. all are important for you. Also, to avoid false high test results for Free T4 and Free T3, you should defer your morning thyroid med dose until after the blood draw.