"She told me that she isn't concerned by it because it's TSH they look at and not T3..."

That's sort of like saying it's the gas tank indicator they look at and not how much of that gas is actually making it to the engine to fuel the car.

T3 is an activated form of T4 and TSH is the pituitary peptide/hormone that tells the thyroid to produce more T4. Thus one can have normal levels of TSH and T4, which would indicate that the pituitary/thyroid glands as well as the axis between them are fine. Yet there could also be low T3 levels, which could indicate a problem with thyroxin utilization. That sort of problem often involves the metabolic axis between the adrenal glands, the pituitary and the liver (where the conversion is made).

Hormones and how they interact with each other is a very complex study. Given that there does seem to be something going on in that department, an appointment with an endocrinologist might be in order. I wouldn't let the brush off about your T3 levels (if it is severely low) that the internist gave you slide because it can produce all of the same symptoms of primary hypothyroidism.

Good luck.
Please keep us informed.

With love, Erika