Without seeing it, I am not sure what happened or if it will look nice when you iron it on.
Could it be that when you printed it, it used too much ink (like printing on the highest quality) and that the ink ran? Your word "bleeding" sounds to me like the ink is running but I do not know.
I typically always print on best and have not had that problem, so I am unsure.
Did your printer print the green neon like on the screen? I have heard others say that their printer will just not print that color - like it will print the green darker or more muted. I think that may be a computer or printer adjustment issue but I am not at all a computer expert.
Hopefully some of this will jog some ideas for you to try.