Welcome to our fourm,
You have an Autoimmune Thyroid Disease called Hashimoto's due to having thyroid antibodies. It is the antibodies which attack your gland untiil you are hypothyroid but treatment is the same. Going gluten-free can help reduce the antibodies.
When we have blood tests taken for our thyroid hormones, it has to be at the very earliest possible, fasting (you can drink water) and you should allow a gap of 24 hours from your last dose of levothyroxine and the test and take afterwards. This allows the TSH to be at its highest (it drops throughout the day) and can mean the difference between getting an increase or doctor reducing levothyroxine. If you didn't follow this procedure your results may be skewed.
Your GP should also check B12, Vit D, iron, ferritin and folate. Deficiencies can also cause clinical symptoms.
After being diagnosed six years ago never take the word 'normal' with regard to blood test results as they are meaningless. Normal for hypothyroid means we have no clinical symptoms and feel well. 'Normal range' with regard to results we need 'optimum' i.e. TSH of 1 or below. Free T3 and Free T3 towards the upper part of the ranges. These are rarely tested but we have private labs which can do.
Always get a print-out of results with the ranges. Labs differ in their machines and ranges can be different too.