Thyroid hormone monitors growth, cell repair, and metabolism in our body. People with thyroid disorders should avoid goitrogens in their diet. Goitrogens are compounds that interfere with the normal functioning of the thyroid gland in our body. Diet plays a major role in the management of thyroid disorders.
There are certain vegetables that contain goitrogens and should be avoided, For example

1. Cruciferous vegetables such as broccoli, cauliflower, Brussels sprouts, cabbage, and kale. They interfere with how the gland uses iodine for producing hormones.
2. Sweet potatoes, cassava are some of the starch plants to be avoided
3. Fruits such as peaches, pears, and strawberries should be avoided as well

Goitrogens are naturally occuring substances that inhibit thyroid production and cause an enlarged thyroid in some individuals. Cruciferous vegetables are known goitrogens and can act as anti-thyroid medication (thyrostatic) which would be self defeating if you were on medication designed to increase your thyroid levels.

Goitrogens are able to disrupt normal thyroid function by inhibiting the body’s ability to use iodine, block the process by which iodine becomes the thyroid hormones thyroxine (T4) and triiodothyronine (T3), inhibit the actual secretion of thyroid hormone, and disrupt the peripheral conversion of T4 to T3.

From All About Goitrogens

Except for a vegan, raw food type eating massive quantities, I believe it would be difficult to inhibit your normal thyroid levels through diet alone, but if you are already low/impaired thyroid and on meds, it could cause unacceptable reductions. People with thyroid cancer are also asked to sometimes stay away from excessive consumption of cruciferous vegetables.

Role of dietary iodine and cruciferous... [Cancer Causes Control. 2010]

I believe that the thyroid supressing effect of cruciferous vegetables is probably over exaggerated on the Internet due to the relatively well known (and illegal) cattle raising hack which is to feed cows goitrogens so they will have lower thyroid levels which in turn will make them fatter. This causes all sorts of problems for the animals including being carcinogenic. With cattle you can feed them exclusively on cruciferous and other thyrostats so the potential for abuse is very high.

Evidence that urinary excretion of thioura... [Food Addit Contam. 2006]
Analysis of antithyroid drug residues in food an... [Bioanalysis. 2012]
Effect of radish (Raphanus sativus Linn.) ... [Indian J Exp Biol. 2006]

*I am not a doctor, but I wanted to try to provide an answer to this question. Please do not use this answer as anything more than a starting point in your research.