Your TSH is so high because your thyroid levels are very low and your pituitary gland is attempting to stimulate your thyroid to produce thyroid hormones. This is a normal pituitary reaction when the thyroid doesn't seem to work.
TSH is just an indicator that thyroid hormone levels aren't adequate and looking at your Free T4 and Free T3, it's obvious that they're both well below the lowest part of the reference ranges. This indicates that your thyroid isn't responding to your pituitary gland's call for more hormones. TSH neither causes nor alleviates symptoms; it's the Free T4 and Free T3 levels we worry about.
No, it's not safe to have thyroid hormone levels this low, which is why your doctor prescribed the thyroid hormones... Typically, it's best to start at a lower dose or replacement thyroid hormones, such as 25 or 50 mcg and work up to a higher dose, but some people do well with starting at a higher dose such as your 100 mcg Thyrosit.
Thyrosit is a T4 only, medication, which means it has only the T4 hormone in it. Your body, then, has to convert the T4 to T3, which is the hormone used by individual cells in the body. It takes 4-6 weeks for this type of medication to reach its full potential in your body.
Please tell us what symptoms if any, you're having. Does the medication seem to be helping you? Because your TSH is quite high and your thyroid hormone levels very log, plus it takes longer for this medication to start working, it may take longer for symptoms to begin improving.