Your GP sounds very good. Some people find that they get their TSH within range (i.e around 4) but their GP won't increase their dose because they are in range! Your GP is right that there is room for an increase. There is no problem if your TSH goes below range because TSH doesn't correspond very well to thyroid hormone levels in hypothyroid patients. She would be better looking at your T4 and T3 levels.
Shaws is right that we really need reference ranges as they do differ from lab to lab, and the best way to know if you are on the right dose is to go by how you feel
As it sounds like you have a good GP, it would be asking for the following tests as they could be contributing to your weight and tiredness;
ferritin (best 70-90 for most people)
vitamin B12 (if below 500mcg, consider supplementing with methylcobalamin form of B12)
Low levels of any of the above could be contributing to your symptoms (as well as needing more thyroxine) so it's a good idea to check them out. They can also affect how well your body is able to use the thyroxine you are taking.
I hope that helps