I have had thyroid problems myself with graves disease and have studied the thyroid gland.
most people in western culture in the UK and USA will not have iodine deficiency since iodine is supplemented in almost all salt and sodium in our diets.
and just a small teaspoon of salt each day is enough iodine for our thyroid glands.
but if you have antibodies against your thyroid gland and low activity of your thyroid gland, you could have something called hashimoto thyroiditis
you need to see an endocrinologist
and get blood testing to check your anti thyroid peroxidase and anti thyroid globulin antibodies
and your thyroid hormone levels of TSH and free T4 and free T3 levels.
taking iodine alone will not make your thyroid make more thyroid hormone.
taking extra iodine will not have a negative impact on your thyroid gland.
make sure you take your thyroxine on an empty stomach with no other medications and no food 30 minutes before and afterwards.