Weight issues are very common with hypothyroidism, because thyroid hormones control metabolism. The first thing I'd have to ask is whether or not your actual thyroid hormone levels are where you need them. Many times, doctors only test TSH and try to balance thyroid hormones, based only on that. That doesn't work, because TSH is a pituitary gland, not a thyroid hormone. If you haven't had Free T3 and Free T4 tested to make sure they are optimal, you need to insist that they be tested. Feel free to check out the Thyroid Disorders Community for more information about balancing thyroid hormones. You can access it via the following link:
Other problems could be insulin resistance or pre-diabetes, which happens when our body doesn't use insulin properly. Insulin is often referred to as the "fat storage hormone" because whatever blood glucose that isn't used for immediate energy is stored, by insulin, as fat. If the cells in your body are resistant to insulin, they won't accept the glucose properly.
Yet a third issue could be Metabolic Syndrome, which is a set of conditions, including high blood pressure, high blood glucose, overweight, especially around the middle, high cholesterol and high triglycerides. Having just one of these conditions doesn't qualify as Metabolic Syndrome, but having the cluster does.
Both insulin resistance and Metabolic Syndrome, are often present with hypothyroidism.
To answer your question... I've had this problem for the past 8 yrs. Last yr, I managed to drop 26 lbs by adding more healthful fats, in the form of coconut and olive oils, butter, avocados, etc to my diet. I went back to full fat dairy, instead of low/no fat, because when they take out the fat, they often add extra sugar to put some taste back in. The fat helps satisfy hunger, without increasing blood glucose, so I ate less overall. I felt less hungry and didn't have the cravings I, typically, had. At the same time, I added lots of fresh or frozen veggies, cut back on fruit (because of the sugar content) and eliminated almost all processed food and breads, crackers, etc. I switched from regular pasta and white rice to whole grain pasta and brown rice. I also baked quite a bit with gluten free flours, though those do not tend to be lower calorie - some are higher fiber...
After losing the 26 lbs, I stalled for some reason and over the course of a year, have gained back 6 lbs that I can't seem to lose again... That could be because I'm not exercising enough - or it could be because I'm craving potato chips... !!
So - first to check is the thyroid hormones, then insulin/blood glucose, then Metabolic Syndrome...