Don't expect me to tout a vigorous exercise program to help you lose weight if you're hypothyroid -- it ain't gonna help, and it might hurt. Now, is that what you wanted to hear or not? ;-)
Start looking into the subject, and you'll find a lot of the old conventionalwisdom. Since hypothyroids find it easy to gain weight, CW dictates anything you do to combat this is a good thing, including those lovely strictures about aerobic exercise and a balanced diet. The more progressive sites might actually recommend strength training and a lower-carb diet. If you're persistent in pushing that "next" button on Google, you'll eventually find a couple of actual evidence-based sources of information.
Like here: i learned that hard exercise "eats through" thyroid hormone like it's going out of style (kinda like eating a high-carb diet). If, like me, you make enough hormone to get you through a good day but are challenged when things don't go just right, this is a recipe for massive discomfort. In another study, the upshot i got was that you can manipulate the lab numbers all you want, but it's not going to improve the body's actual performance.
This is not to say that i don't believe in ANY exercise, but i now find the argument weak that we believed in so long -- that exercise revs your metabolism, and that a revved metabolism results in weight loss. Wooo's post about that convinced me ... along with my personal experience.
What exercise is best for seems to be the chance to get outdoors, breathe some fresh air, and move at a constant pace for a half-hour or so, giving all my muscles a chance to extend and contract, not just the ones between my ears. ;-) More importantly to those of us who find weight-loss challenging, i believe, is encouraging mitochondrial health and development, which is what the tabatas do.
My experience says that a good brisk walk in nice weather is both pleasant and helpful; it improves the wind (no, not THAT kind of wind) and i find it also dampens appetite. The little bit of tabata-sprinting i've done on the stationary bike seems constructive, too, but what feels best is about three sets of 15-20 seconds of hard pedaling with a couple minutes slow movement in between -- certainly not the full minute of all-out effort in one go, that some sources recommend.
I don't know about you, but i consider my findings rather encouraging! Even if i were convinced that hard aerobic exercise was going to be beneficial ... well, it just would NOT happen. KNOWING that working that much is actually problematic makes me face a "workout session" a lot more cheerfully. I don't have to feel guilty -- as if i ever would -- under the misapprehension that self-torture is somehow good for one.