Besides genetics, what determines the chance of a solid thyroid nodule being cancerous? I have a solid nodule that has grown fast and normal TSH level

Cancer risk. Risk of cancer is higher in patients < 20 and > 70, and in males. Concern for cancer higher if fixed (non-mobile), fast growing, lymph nodes in the neck, firm/hard, history of neck radiation, problems swallowing/talking/breathing. A TSH that is normal is actually more concerning as well. Typically if > 1cm and normal tsh, biopsy is performed. Especially if any other risk factors present.
Risk. Agree with dr. Wagner's answer. Thyroid nodules are common, however, personal history of excess radiation to the neck or family history of thyroid cancer are two strong risk factors. Depending on the original size, rapidly growing solid nodule need to be evaluated with a biopsy. Tsh can be normal in thyroid cancer. When in doubt, biopsy it.
Fast growing nodule. Most people with thyroid nodules have normal tsh. Less than 1% of nodules cause hyperthyroidism or thyrotoxicosis. Five percent (5%) of all palpable thyroid nodules are cancerous. Therefore, a fast growing solid nodule with a normal TSH should be biopsied to rule out malignancy. Risks : male, extreme age (<20 or >65), fast growing nodule, symptoms, history of radiation to head/neck, family history.