Well if you have half of your gland only, it may not be able to make enough hormones. In order to force it to work harder, your pituary releases Tsh, thyroud stimulating hormone. If your tsh is rising, it shows that the thyroid is having to be coaxed into producing more hormone.
The rising t4 shows that the half a thyroid you have, is responding to the rising tsh by producing more hormones. so, far, so good.
in the uk, the docs can leave you until you are flogging the poor thyroid to death in order to make it work. They believe that a tsh of 10 is acceptable, but in most other countries, a tsh of 3 shows that the thyroid needs a helping hand in the form of thyroid hormones.
Your ideal situation is a tsh of around 1 and free t4 well in range. Make the doctor keep a very close eye on you and if the tsh is still rising but the free t4 is static, or falling, insist on replacement thyroid hormones. (Levothyroxine) or a referral to a competent endocrinologist.