Most people on levo need low TSH around (or even slightly below) 1 to be adequately treated.
If they have not been done ......Suggest you ask GP to check levels of vitamin d, b12, folate and ferratin. These all need to at VERY GOOD (not just average) levels for thyroid hormones (our own or replacement ones) to work in our cells
Also have you had thyroid antibodies checked? There are two sorts TPO Ab and TG Ab. (Thyroid peroxidase and thyroglobulin) Both need checking, if either, or both are high this means autoimmune thyroid - called Hashimoto's the most common cause in UK of being hypo.
(NHS rarely checks TPO and almost never checks TG. NHS believes it is impossible to have negative TPO and raised TG. It's rare, but not impossible, there are a few members on here that have this.)
ALWAYS Make sure you get the actual figures from tests (including ranges - figures in brackets). You are entitled to copies of your own results. Some surgeries make nominal charge for printing out. Alternatively you can now ask for online access to your own medical records. Though not all surgeries can do this yet, or may not have blood test results available yet online.
When you get results suggest you make a new post on here and members can offer advice on any vitamin supplements needed
If you can not get GP to do these tests, then like many of us, you can get them done privately
Blue Horizon - Thyroid plus eleven tests all these. (£99)
This is an easy to do fingerprick test you do at home, post back and they email results to you couple of days later.
Usual advice on ALL thyroid tests, (home one or on NHS) is to do early in morning, ideally before 9am. No food or drink beforehand (other than water) If you are taking Levo, then don't take it in 24 hours before (take straight after). This way your tests are always consistent, and it will show highest TSH, and as this is mainly all the medics decide dose on, best idea is to keep result as high as possible
If you have Hashimoto's then you may find adopting 100% gluten free diet can help reduce symptoms, and lower antibodies too.
You do not need to have ANY obvious gut issues, to still have poor nutrient absorption or low stomach acid or gluten intolerance.
Best advice is to read as much as you can. Vitamin and minerals levels are very important, but standard NHS thinking, doesn't at the moment seem to recognise this. You will see, time and time again on here lots of information and advice about importance of good levels of B12, folate, ferritin and vitamin D, low stomach acid, leaky gut and gluten connection to autoimmune Hashimoto's (& Grave's) too.