Hello! Your daughter may need more testing done. The doctor needs to test the thyroid antibodies: Thyroid peroxidase antibody and anti thyroglobulin antibody. Repeat TSH and Free T4. With thryoid testing if TSH is high and Free T4 is low=hypothryoidism: signs and symptoms: feeling cold, constipation, tired, etc. If the Free T4 is high and the TSH is low.
If your daughter is hypothryoid she probably has Hashimoto's thyroiditis. This is often treated when your daughter's free T4 is real low and TSH is real high. Treatment: Synthroid. If antibodies are positive and free T4 and TSH are fine, then your child's thryoid function should be checked routinely.
It sounds like she may be hypothyroid, though that is normally associated with high TSH. Constipation is a symptom of hypothyroidism. You may want to post your question on the thyroid forum here - there are many experienced people there. What did the doctor say?