High TSH when you're not on Iodine most often means that your body is desperately looking for iodine. It can also enlarge in this situation to make a greater surface to get as much Iodine as it can.
High TSH when you are on Iodine means that your body is making NISes. A NIS is a mechanism that gets iodine into the cell. High TSH nudges the body into making more NISes.
When the NISes are back and iodine is still sufficient, TSH will drop, because the current flow of iodine into the cell is satisfactory.