Yes it could be better if you still feel symptoms. It could be at the bottom of the range and you will be fine if not feel much better.
The GP has overlooked the fact that In a normal thyroid some t3 is made directly (about 20% of the total). The rest is converted in tissues and the liver from t4. T3 is difficult to test as it varies during the day with a short half life, t4 has a long half life and is a bit more stable during the day. T4 is easier to administer but therefore we need a little more to accommodate the lack of our own t3 (t3 is what is actually used by the body, as well as t2 and t1, t4 is really a pro hormone).
To accommodate the conversion it helps if ferritin, vit d etc are all really good too.
Therefore most patients need a complete suppression of tsh to feel well and have a good supply of enough t3.
This is clearly explained in the BMA book understanding thyroid disorders by Dr A Toft. We should be aiming for the lower ref range - 0.3 is fine if you feel ok at that level and t4 in upper, around 20-22 (I think). You should take the pill after a blood test and have the tests roughly the same time in the morning. (An endocrinologist told me this)
I found buying the book (recommended here!) and showing my GP really helpful.
Toft even says that ideally we need a treatment that includes t4 and t3, it is available but can be tricky as the t3 has to be taken frequently. Some find they get too buzzy on it as well.
I'm finding that, after slowly getting my ferritin and vit d really good, and getting the tsh to about 0.5, and slowly building up my strength etc again I'm well like this.,