FYI, I typically supplement with Iodine (Lugol's) a few times per year at 30-50 mg/day for about two weeks. I was supplementing during a recent physical exam and TSH came back at 5.8. Stopping in anticipation of a one month follow up, TSH was 2.4 only 30 days later. T4 and T3 are normal.
Clearly, TSH can be strongly influenced by short-term iodine supplementation. Some claim TSH comes back down after long-term supplementation, but I have no labs to confirm that.