If she won't turn at the trot, chances are she does something similar at the walk. Tiny problems generally get worse the more speed you add. Go back to the walk and get perfect control there. Forwards, backwards, left and right. Build a "whoa" button. Get control of her shoulders, ribcage, and hindquarters. From the situation you described, teaching a one rein stop is also in order. You can teach this in a halter or in a bridle, from the ground and from the saddle. All of the basic control exercises can be introduced from the ground and translate into ridden work.
As far as the bit issue, if you bought her, she's your horse, not the boarder's or the first trainer's. I personally would start letting her wear a snaffle bit headstall with the halter when you lunge and ride, let her get used to the bit in her mouth. When she's comfortable with that (no head tossing, easily accepts bitting, etc.) you can introduce bit pressure and ask her to respond to rein cues.
The best thing I can tell you is, if it's not good at the walk, the trot will be a real nightmare. Same for trot to canter.
Incidentally, how high is your roundpen? Except for watching Parelli's on TV, the shortest one I've ever seen is 5 1/2 feet. That's a heck of a jump! Kudos for staying on!
A stubborn horse walks behind you, an impatient one in front of you, but a noble companion walks beside you ~ Unknown