Once hypothyroid and on levothyroxine (or any other thyroid hormone replacements) it is for life. It can be fatal if we stop taking it. That's the reason we pay for no other prescriptions for any other illnesses in the UK>
I suggest you get a brand new blood test and if you can afford it a Private One which will do all of the ones you need. However, if money is tight the GP should do some and you may be able to get Free T4 and Free T3 (once should be enough so that you know how you are converting levothyroxine.
Levothyroxine is T4. T4 is inactive and has to convert to T3. T3 is the only active thyroid hormone used by the receptor cells.
Ask GP to Test, TSH, T4, T3, Free T4, Free T3 and antibodies (if you've note had antibodies tested before. B12, Vit D, iron, ferritin and folate.
The lab or GP may not do all of the thyroid hormones but he should do all of the Vitamins/minerals.
The test should be at the very earliest, fasting (you can drink water) and allow a gap of 24 hours between your last dose of levothyroxine and the test and take it afterwards. This helps give the highest TSH as that's all doctors appear to take notice of.
Get a print-out of the results with the ranges. Ranges are very important as labs differ in their machines and the ranges differ. It makes it easier for members to comment.
Tick off the symptoms you have and tell GP that you've taken advice from the NHS Choices for information and advice and show him how many symptoms you have. You should have none if there's sufficient T3 in your receptor cells. We have billions.