Lithium is known to destroy thyroid function.
I also take lithium for bipolar disorder and have been on it for 5 1/2 years. I believe it has been affecting my thyroid for quite some time now. I developed my first thyroid problem, hyperthyroidism, after taking it about a year. They treated the thyroid problem with thyroid hormone supressing drugs, but they had me on too high a dosage. I ended up becoming hypothyroid and was eventually taken off all thyroid medication. I was fine for a couple years, then 2 years ago I became hyperthyroid again. I went back on the thyroid medication, but on a lower dosage. In the past 6 months I've noticed a huge change in myself and am experiencing many symptoms of being hypothyroid again. My doctor claims my test results are "normal", but I'm not convinced. I feel terrible all the time. I believe the lithium is continuing to cause my thyroid instability. I don't recommend you stop the lithium, but you should educate yourself as much as possible about thyroid disorder and be proactive with all your doctors about addressing any issues you have. If you're not getting the response/care you want, then push harder or find new doctors.
It is not unusual for people with bipolar disorder to also have autoimmune thyroid disorder.
http://www.psycheducation.org/thyroid/introduction.htm
Thyroid problems do also tend to be hereditary, I've been told. Your TSH level is significantly high and would normally be treated with thyroid medication. The doctor will be testing you further to see the effect of the med. At that time you should strongly suggest that your free T3 and free T4 be tested also. Free T3 is the most active component, and largely regulates your metabolism. For best results, members here have found it best to have free T3 in the upper third of its range and TSH in the low end of its range. Don't let the doctor medicate you strictly on the basis of TSH. That doesn't usually work very well.