Your trainer sounds rather unprofessional.
A number of things could be contributing to your problem, but don't feel bad about it.
Happens to everyone.
Your trainer probably wanted to challenge you with this green horse. Does this horse normally have such an issue with forward movement and transitions? Can you make this horse walk fast versus slow? Do you have a lot of leg on your horse even while you're not asking for anything?
If your trainer can make this horse trot perfectly on cue, and assuming this horse is not in pain (coughcough), I'd be inclined to think that (1) you are tense, and (2) maybe your cue is a little muddled. Are you blocking the horse's forward movement with your hands? Is your cue very definitive, and do you maybe have a preparation cue (like exhaling, or something verbal)?
When I ask for a trot, I first ask my horse for a swinging, energetic walk where his hind end is as engaged as I can get it (still working on that...
). I might do this by "pushing" his barrel side to side with one leg and then the other to ask him for more energy. This will make the transition easier. If your horse is lazing along on his forehand and you suddenly ask for a trot, he isn't prepared and will probably move along at an agitated, faster walk or shuffle into some semblance of a sloppy trot.
When my horse is swinging along at the walk, I will start to think about my trot cue. Before the cue, I will count in my head "one, two, three" or say outloud "aaand trot," and give my cue which is a very distinct squeeze with both legs. If you need more refinement, you can use the edge of your heel or a tiny pair of nub spurs (if you are capable of riding with them) to give that squeeze some more specificity so that your horse can say "okay, this is different from what I've been feeling all along."
If your horse doesn't listen after this point, then your instructor absolutely needs to weigh-in. But keep in mind... transitions are all about preparation! If your horse is prepared and you are relaxed, that transition can be smooth and timely.