Yep...an underactive thyroid results in a slow-down of all bodily processes...a sluggish metabolism will effect everything. If you know that you are low thyroid you might want to see a doctor about using thyroid therapy...Armour is a natural option. Low thyroid can also cause Depression and a build-up of toxins since the detoxification process is also effected. And yes...your thyroid function may improve with cleansing and healthy diet...but you're right about it being a circular relationship and supporting your thyroid will also assist with the clean-up and improved digestion and assimilation.
Also add kelp to your diet...it helps support the thyroid and provides a natural supply of iodine.