There's no reason to supplement with thyroid meds while using GH unless levels are out of range which won't happen with most typically recommended GH protocols much for the average person.
The negative feedback loop (hormone somatostatin) that inhibits GH release because of high levels from the exogenous GH also happens to inhibit TSH release, so there will be some effect on thyroid levels to a degree. Most of the time it's nothing to worry about.
The whole idea about T4 to T3 conversion making GH more anabolic is a made up theory from what I have seen. No real scientific data has been presented to validate that claim. I call bullshit on that theory, especially since it came from untrusted sources in my opinion.
With that said, I like to add a little T3 or T4 to get my levels into the end of the hyperthyroid range. This I feel is ideal for burning fat while minimizing any catabolic effects. Going further hyperthyroid out of range isn't recommended as it will burn muscle. I aim for a 0.4-0.6 TSH level.