Is there synthetic tsh?

No: Tsh is produced in the anterior pituitary gland in the brain. Its function is to regulate thyroid hormone production in the thyroid gland itself. The TSH test is just to monitor that feedback loop. If there is too little thyroid hormone, then that is given exogenously because that has a physiologic function. Tsh does not. If the pituitary is treated if it's abnormally producing extra tsh.

Yes: It is called thyrogen. It is used for testing after thyroidectomy for cancer. It allows for testing for rai scan and thyroglobulin without thyroxine withdrawal.