I would guess that you are taking levothyroxine and in this situation, there is an alternative which is another thyroid hormone called T3....However....i do not feel that this is a consideration in your case.
Providing that your blood tests indicate that you are on the correct dose of thyroxine, then i can't see a direct relationship between your breathlessness and your thyroid medication.
If this breathlessness continues, then see your GP as it may well be a manifestation of other problems.