The main thing you must do is find out your tsh from the Gp and work with them to maximise your levels.
Most need to be around 1 or below, even below the bottom range (usually 0.4ish) to feel well.
It's mostly water retention which is then lost as tsh gets lower. You become more active and naturally loose a bit of weight too and will be more able to properly exercise.
In my experience of over 20 years of it though, strength training is really important for a range of reasons with hypothyroidism rather than focussing just on loosing weight.
This is the most comprehensive diet guide. The btf are the charitable wing of the British thyroid association (of consultant endocrinologists) and their guides are worked on with consultants and gps. www.btf-thyroid.org/information/108-thyroid-and-diet-factsheet
I know a nutritional research scientist who researches this area and told me that while iodine isn't as important when your on thyroxine, still keep it in your diet via dairy and white fish, selenium too (so Brazil nuts!) She also told me iron levels are important (ferritin over 70, iron levels can affected by hypothyroidism, something to do with the bone marrow metabolism ) which I'd agree with from experience so eat a diet rich in iron with orange juice, avoid tea and milk near those meals. (Tea and dairy reduce absorption)
They generally say to make sure your b12, folate, ferritin and vit d is good, mainly as deficiencies can mimic the tiredness and achiness of hypothyroidism.
The gluten thing isn't true. There's a higher incidence of people with coeliac disease who go on to develop auto immune hypothyroidism which is where that comes from. Some think if you have hashimotos that avoiding gluten reduces the antibodies; I've never found a credible source for this after much searching.