Okay, so in view of his symptoms you mentioned and his Free T4 being so low, and the TSH being higher than would be expected, I suggest that you have a further discussion with your doctor and ask that he be tested further. Specifically, from his TSH I suggest that he should be tested for the autoimmune antibodies associated with Hashimoto's Thyroiditis. The tests required are TPO ab and TG ab. In addition he was only tested for Total T3, not Free T3. The Total result reflects all his serum T3. Most of that is bound to protein and thus biologically inactive. In view of the importance of Free T3, you should make sure they always test for both Free T3 and Free T4 every time he goes for tests. In addition it would be a good idea to do a morning serum cortisol test, since optimal cortisol is so important. In addition since hypothyroidism so frequently results in deficiency in Vitamin D, B12 and ferritin, those should also be tested.
Most importantly you are going to need to find a good thyroid doctor for your son. That does not necessarily mean an Endo. Too frequently they seem to only pay attention to TSH for both diagnosis and treatment. That does not work. A good thyroid doctor will treat clinically, by testing and adjusting Free T4 and Free T3 as needed to relieve symptoms. Going forward, always keep in mind the following conclusion from an excellent recent scientific paper, "Hypothyroid symptom relief was associated with both a T4 dose giving TSH-suppression below the lower reference limit and FT3 elevated further into the upper half of its reference range." And frequently achieving adequate Free T3 levels requires the addition of a source of T3 to meds. If you will give us your location, perhaps we can suggest doctors who have been recommended by other thyroid patients in your area.
In view of his growth problem it would also be a good idea to test him for testosterone levels. I am sending you a PM with some info in that area. To access, just click on your name and then from your personal page click on messages.
So what ranges did the PCP suggest using to evaluate the test results? Pediatric ranges tend to be somewhat higher than adult. You can notice this is info like the following:
https://www.researchgate.net/publication/23499982_Pediatric_reference_intervals_for_thyroid_hormone_levels_from_birth_to_adulthood_A_retrospective_study
When looking at pediatric thyroid test results and they are in the lower part of their range, I don't think the PCP really understood that those results are lower than optimal for many people. Your son has several symptoms that occur frequently with hypothyroidism. As I mentioned symptoms should be the first consideration, followed by evaluation of thyroid test results. I cannot assess his test results without at least knowing the measurement units for them. Is the FT4 in ng/dl, or pmol/L? Is the FT3 shown in pg/ml or pmol/L?
There is much to be discussed, but first please post the reference ranges shown on the lab report for those test results. Also, even more important than thyroid related test results are symptoms. Please tell us about any other symptoms you have noticed.