Can a lack of thyroid make you prone to higher estrogen, i.e. does thyroid help balance estrogen, or get rid of excess estrogen?

I know high estrogen in itself can suppress thyroid, but if you took the high estrogen out the equation (ie levels were balanced) and something else caused hypothyoidism, could the lack of thyroid then make you prone to high estrogen?

Indirectly yes. A lack of thyroid hormones can increase body fat.  Fat cells also generate estrogen. Therefore, the more body fat present the higher the body's estrogen levels become.