T-4 will help to offset the lowering of the thyroid.
Another factor is that the body has to get used to the amount of exo-GH and this takes time.
Allthough I know a guy who has been on vey high doses of GH (professional) and whenever I see him he is always yawning and sleepy. So I think that past certain high doses you will just keep feeling tired and sleepy (a good example of this possible effect is one of Jay Cutlers vids where he talks about supplementation but is so sleepy he can barely talk).