Here is what the National Institute of Health says:
“In total, 14 trials (thyroid function was not the primary health outcome in any trial) were identified in which the effects of soy foods or isoflavones on at least one measure of thyroid function was assessed in presumably healthy subjects; eight involved women only, four involved men, and two both men and women. With only one exception, either no effects or only very modest changes were noted in these trials. Thus, collectively the findings provide little evidence that in euthyroid, iodine-replete individuals, soy foods, or isoflavones adversely affect thyroid function. In contrast, some evidence suggests that soy foods, by inhibiting absorption, may increase the dose of thyroid hormone required by hypothyroid patients. However, hypothyroid adults need not avoid soy foods. In addition, there remains a theoretical concern based on in vitro and animal data that in individuals with compromised thyroid function and/or whose iodine intake is marginal soy foods may increase risk of developing clinical hypothyroidism. Therefore, it is important for soy food consumers to make sure their intake of iodine is adequate.”