There are a few causes of low thyroid, one of them can be low Iodine. Including Celtic sea salt into your diet can be very beneficial! Or just putting a few drops of Lugo's Iodine into the water can help a lot! Another one is Copper toxicity, and anyone on the pill probably has it... The best way to fix that is to supplement with Zinc (try biglycinate form) which replaces copper. 50 mg per day should be a good amount. Zinc also helps with digestion, immune system and it is basically in every chemical reaction in our body - wonderful mineral, not gonna hurt anyone!
Focus on foods that are not processed and contain iodine (sea food) and that are high in Zinc (lamb and sea food)
About goitrogens in cruciferous vegetables - I wouldn't recommend avoiding them all together since they are very healthy, but you must cook them well!
I hope you find this useful! I have a blog if you want more nutrition advice (www dot nadyapeche dot com)
Good luck!