if this were my dog from what you are describing i would take a step back and on the next time for flea protection i would use revolution as it is for flea, heart worm and tick and it is used to rule out scabies (sacroptic mange) and much prefer that to toxic dips
and i would do natural balance venison and sweet potato for 12 weeks there is a canned version as well to pill with. I would do this for 3 months
also if the dog was not tested for thyroid I would get that panel done as i am freaking out reading this not knowing if a thyroid panel was done and a vet giving thyroid meds as that is just nuts to me. You can have the thyroid panel done through michigan state or dr jean dodds as we see dr jean dodds and she is an expert in blood work.
I personally think it has something to do with scabies or food if steroids are not giving relief - i do not suspect demodex as demodex is not that itchy but scabies is.