It does make you taller, at least until your growth plates have fused. For example, somatropin, a synthetic growth hormone, is sometimes given as an injection to children who are significantly smaller than expected to help them grow taller. Once their growth plates have fused, however, a person's height is set, and so there's no way to become taller without breaking bones. If a grown adult took somatropin, their body would build muscle instead of making them taller.

ETA: excess growth hormone can lead to major health issues, which is why it's not typically prescribed for regular-height people to become a bit taller. Somatropin, for example, increases your risk of brain and skin cancers.