Most fruits and vegetables contain some iodine, but a few are higher in iodine than others.
Watercress, onions and strawberries are good sources.
But it really depends on the soil they have been grown in. If the soil is iodine-rich, the fruits and vegetables will also be iodine-rich. Some areas far away from the sea have iodine-poor soil.
To know where the soil is iodine-rich and where it is not is not easy for us who are not scientists, but they say that in the middle of the US the soil is quite low on iodine as well as in the South West. In Europe there are also areas in the middle of Europe, like for example Switzerland that has soil low in iodine.
A tip would be to eat large amounts of vegetables and fruits from different parts of the country.
Another natural source of iodine is seaweed. More specifically kelp. Ideally, buy Japanese kelp (kelp from Japan), but if you cannot find that in the US, you can buy any other variety, for example kelp from California.
In Bartrams book, "Natures Plan For Health", there is a Thyroid gland vegetable juice therapy recipe:
Spinach
Watercress
Lettuce
Radish
The juice should be juiced and of course raw. It would be best if you would get yourself a juicer to be able to make yourself this juice.
Dr. Bartram recommends a glass or two a day of this fresh raw juice. But I would say at least two glasses per day for you, since you are already diagnosed with a thyroid goiter problem.
And Dr. Lynne August says we need 150 micrograms of iodine every day. So, how much raw kale should we eat if we want to get all our iodine from raw kale?
Eating kelp and nori would probably be an easier way getting your iodine, where only one gram contains anywhere from 16 micrograms to 8000 micrograms! The only problem there is you might get too much iodine if you eat large amounts, so just eat small amounts of seaweed a day plus the veggies and the juice and you should be fine!
Wild plants normally have higher mineral contents than organic or conventional fruits and vegetables, because their roots go really deep into the ground.
So you could for example try eating some dandelion leaves each day too!
I know they taste bitter, but maybe you could chop them up and them to your salad?