If the doctor disputes any of the info in the link, just suggest that there is extensive scientific evidence for everything in the paper, and that he will not find conclusive evidence to the contrary. The doctor is following a standard of practice that is outdated and is not adequate to eliminate hypothyroid symptoms for the patient. There is lots of info in the paper that proves that the FT4 and FT3 ranges are far too broad, and that every person can have different levels at which they feel best. Just being anywhere within the erroneous range limits does not work for many people. And if the doctor is one that mostly pays attention to TSH, there is references to scientific studies showing that TSH is usually suppressed when taking thyroid med adequate to relieve symptoms. Also don't overlook the importance of Vitamin D, B12 and ferritin.
A good thyroid doctor will treat a hypothyroid patient clinically, by testing and adjusting Free T4 and Free T# as needed to relieve hypo symptoms, without being influenced by resultant TSH levels. Symptom relief should be all important, not just test results, and especially not TSH results when taking thyroid med. Thyroid med dosage should be raised enough to relieve signs/symptoms of hypothyroidism without creating any signs/symptoms of hyperthyroidism. Many of us have found that this required Free T4 at least at the middle of its range, and Free T3 in the upper third of its range, or as needed to relieve symptoms.
An excellent thyroid doctor says that in tests done about 24 to 28 hrs after their last daily dose, most people on adequate NDT therapy have a suppressed TSH. They usually have FT4 levels that are 1 to 1.3ng/dL, and free T3 levels that are rather high in the range or even slightly above the range. The higher FT3 level compensates for the lower FT4 levels on NDT. These patients have no symptoms or signs of hyperthyroidism--if such occur the dose is reduced." So two things of note here. One is the levels frequently required to relieve symptoms. Two is that the thyroid med should be delayed until after blood draw for tests, in order to avoid false high results. Did you take your morning dosage of the med before the blood draw? Do you think your doctor would be agreeable to treat clinically, as described?
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. If you need to convince your doctor to treat clinically, you might also find the link useful that way.
http://www.thyroiduk.org.uk/tuk/TUK_PDFs/diagnosis_and_treatment_of_hypothyroidism_issue_1.pdf
Since hypothyroid patients are so frequently deficient in Vitamin D, B12 and ferritin, those need to be tested and supplemented 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.
Lab results and associated reference ranges vary from lab to lab. So please post the reference ranges shown on the lab report for those results.
Even more important than lab results are symptoms. Please tell us about any symptoms you have.
Also when first diagnosed as hypothyroid, what was the identified cause? Was it Hashi's?