Gimel has pointed out some crucial tests to have done. Good for you that you are on top of this for your daughter and not chauking it all up to pre-puberty symptoms.
My now 20 year old daughter was diagnosed with Grave's disease at 17, but most likely she was suffering some symptoms of both hyper- and hypothyroidism at least since she went through puberty. It now looks like she SHOULD have been tested for Hashimoto's, since she may be positive for that as well. When you are asking your daughter's physician about tests, you might want to add TSI (the test for Grave's) to the list. Both Grave's and Hashimoto's are autoimmune conditions: Grave's can make the thyroid overproduce, and Hashimoto's can slowly destroy the thyroid.
I hope your daughter is able to be seen by a pedicatric endocrinologist. This kind of doctor should be well versed in running all of thse tests, as well as ordering any scans if your daughter's thyroid is enlarged in any way. Wish you well!
How high was the TSH? Has she been tested for FT3 and FT4? TSH is a pituitary hormone that is affected by so many variables that it is not a reliable diagnostic for thyroid problems. Of much more importance is the actual, biologically active thyroid hormones, free T3 and free T4. FT3 is most important because it is four times as active as FT4, plus FT3 correlates best with hypo symptoms, while TSH does not correlate well at all. Unfortunately most doctors seem to predominantly rely on TSH.
I would suggest that she needs to be tested also for the thyroid antibodies, TPO ab and TG ab, to determine if Hashimoto's Thyroiditis is involved.
Going forward, remember that the best way to treat a hypo patient is to test and adjust levels of FT3 and FT4 with whatever type of medication is necessary to relieve symptoms, without being constrained by resultant TSH levels. Symptom relief should be the objective of treatment, not adjusting TSH level. If her doctor has a problem with treating her symptoms in this way, then you will need to find a good thyroid doctor that will do so.