Grass landings for the uninitiated normally used to tarmac can be a bit of an eyeopener. Its bumpy, its noisy and the plane never seems to stop flying. It requires discipline and good technique to do a good grass landing (or any hard surface for that matter) and even more so for touch and go's
From your profile I assume you fly from Biggin Hill normally and I think I can guess which grass airfield you visited. I would have thought that one of the major problems here is the difference in perspective from landing at a wide long flat runway to landing on a shorter less clearly defined grass runway.
Sounds like a classic case of a bounce followed by porpoising from what you describe and really your only option should be to go-around from the first bounce, especially as you are already halfway down the runway (from your description). In fact, if you were still doing 80 knots on final with any flap setting you should have gone around much earlier, PA28s stall at 50ish knots with full flap so you're going to float a LONG way at 80 as well as having a very flat or even slight nose down attitude. 65-70 would have been a better choice depending on flap choice which from my book of many near disasters should have been 3 stages.
But since you only have 13 hours you should not be getting guidance from this forum, your instructor should have provided a thorough de-brief on what went wrong, followed by a demonstration of the correct technique, followed by you practising more touch and go's. Instead it sounds like you embarrassed him and he cut your detail short. You should also insist that you are shown Go-Arounds on your next lesson because there is a lot more to it than simply slamming the throttle open and hauling back on the stick.