First please tell us the symptoms you are having! Are you feeling hyper, or hypo or neither?
Secondly, can you verify that the T4 test was in fact for FREE T4, and not Total T4. the range is a bit odd for Free T4, but perhaps the units are the issue.
Finally, you are missing the most important of all tests. The Free T3 test. But if the lab and doctor were fighting you on even getting Free T4 done, and the Dr thinks you are "normal". Then you probably are going to have a very difficult time getting re-tested again for both Free T4 and Free T3.
Reason why I think free T3 is important, is because a lot of the time the pituitary "sees" or senses the body's free T3 level. And if it senses that it is high enough, it will cut production (lower) TSH.
You are only at 16% of the range of Free T4. With low TSH, the thyroid production would be lower, and thus a lower FT4 is not toatally unexpected.
Do you have a diagnosed thyroid condition and are you currently taking replacement thyroid hormone medication? If so, what medication are you taking and what dosage? How long have you been on this dosage?
I agree that you need to have Free T3 tested, along with the TSH and Free T4. If the lab only ran TSH with the idea that Free T4 would be done if TSH was out of range, that could have to do with the doctor's order... in other words, he may have ordered TSH w/reflex to Free T4, which is exactly what you said - they do TSH and if it's out of range, they go on to do Free T4, but if TSH is in range, they don't go any further.
As flyingfool said, it you have to fight to even get Free T4 done, I'd suggest you find a different doctor because it's unlikly that this is going to be willing to order Free T3 if he even knows what it is.
All of that said, if you aren't on a replacement thyroid hormone, it's possible you could have a condition called Central (or secondary) hypothyroidism. This is when the thyroid works fine, but there's a pituitary problem in which there isn't enough TSH to properly stimulate the thyroid to produce adequate thyroid hormones.
A list of your symptoms would be helpful.
Lukeman: If you live in the USA, you can go to a lab without a request from your doctor and pay to get tests done. You can find them online. For instance I just found one that offers a group of tests labeled "Optimal Thyroid Health Profile Blood Test" which includes Total T4, Free T4, Total T3, free T3, TSH, Thyroid Peroxidase (TPO) Antibodies, Reverse T3 for $215 USD today. You could add on the Thyroglobulin Antibody Blood Test. Do the tests, then give a copy to your doctor if you want to. That's what I would do...
It is not an option in most provinces. However it's easy if you are determined, just drive to the US or do it next time you fly to Florida or Pheonix. There are large US labs with offices in most states; they may have a central lab along with sites in many cities where they take the blood sample.
Before you go, set up an account online, select your tests, pay with credit card, print off a receipt/requisition. Give them a US address, the address of a hotel near the lab will do. Take the req to the lab and they poke your arm. In a few days they post your results to your account which you can access with a password. Print off the results. I've done it several times; works good.
It's harder to get proper thyroid testing in Canada, from what we've seen from previous members, but it can be done. You just have to find the right doctor.
It's also customary for labs there to decide what tests to run, based on the health system guidelines, but that, too, can be over-ridden, based on your doctor. Some people have also paid privately for them, if that's an option.
Since you don't have a diagnosed thyroid condition, I'd suggest you find a doctor that's willing to look into possible Central hypothyroidism - the possibility of a pituitary issue - since you have, both, low TSH and low Free T4.
You have quite a few symptoms of hypothyroidism and whatever the cause of hypothyroidism, it needs to be treated.
Have you been tested for Vitamin B-12 or Ferritin? B-12 deficiency can the cause or contribute to brain fog/inability to concentrate issue, as well as severe exhaustion. Iron deficiency can cause hair loss. In addition, both are needed for production and proper metabolism of thyroid hormones. You might also want to be tested for vitamin D, as it's also necessary for proper metabolism of thyroid hormones.
These tests would be helpful in addition to the thyroid (Free T4 and Free T3 tests).
Your labs are almost identical to what my sister's labs were when I asked her to get tested and was, to no surprise, diagnosed with Hashimoto's thyroiditis with both thyroid peroxidase antibodies (TPOAb) and thyroglobulin antibodies (ThAb) high.