The carbs found in most green leafy veggies are negligible, so eat up lots of those! I prefer to get the majority of my carbs from fruit & veggies sources. I avoid most starchy carb sources (breads, potatoes, rice, pasta) but if I do eat those I make it a very small amount and make sure to combine it with a protein and fat. That seems to minimize the blood sugar spike.
Carbs are not the enemy, but over eating the processed carbs is unhealthy (ie: if it's made with flours and/or sugars). Your thyroid needs about 25 to 30 grams of carbs per day to operate optimally, so that's about the lowest I would go.