Quite some time ago, there was a lot of noise in the paleo blogosphere about low-carbohydrate diets causing a drop in thyroid function. I was puzzled by it, as i feel SO much better on LC. Surely if it really DID lower function, i'd be the first to notice it! The high-carb afficionados especially made a big deal about it.
Why do they insist so ardently that their way is the only way? Because they hate eating that way themselves, and they don't want others to benefit from it, what? ...Makes no sense to me.
Upon reading more from a range of sources, it turns out that ANY decent loss of weight makes the thyroid slow down -- LC is just more efficient for fat loss than most others. It's all about energy availability in the diet -- your body is slowing down to protect you from the famine it thinks is coming. From a subjective point of view, i didn't notice feeling worse because it obviously wasn't slowing down MUCH. It may have been statistically significant, but those numbers they use are often arbitrarily chosen. (Read Chris Masterjohn on statistics....)
I suspect that the lack of slowdown in MY body is all about the FIAF and the ghrelin. One thing i DON'T feel on very-low-carb is draggy, dopey and dim -- i have to eat a "normal" amount of carbs to feel that crummy.
So if you're making good progress on LC but are worried about what it might be doing to your thyroid function, monitor yourself for energy levels and symptoms -- but remember that a lot of thyroid symptoms are tied up with nutrient deficiencies ... which may be causing your poor thyroid function in the first place!