You're gotten some really great advice already. Here's my $0.02:
Unfortunately I've seen that a lot in experienced lesson horses who've figured out that they're bigger than the kids riding them and who don't get ridden much by adults who "lay down the law", so to speak.
It's one of those hard situations. What it sounds like Jessie most likely really needs is for someone to come along and really tell her that "stop" means "STOP!!!". That will probably entail really shortening up the reins and hauling on her mouth.
HOWEVER, since your daughter is taking lessons, she's really not the person who should be doing that to Jessie. It's not your daughter's job to retrain this horse, it's the instructor/trainer's job. For me (I teach lessons as well), if a child started disciplining my horse without my permission, they'd be released from my lesson program immediately.
For myself, I make sure to ride my mare very regularly and if she's shown any inkling of any issue in any of her lessons during the week, she's drilled on the issue until it's a non-issue. Sometimes I've even had to cancel lessons because she's being a brat but for me, the confidence of my kids is more important than the money I might lose cancelling.
In any case, I've ridden at my share of barns where the lesson horses are just ridden by lesson kids and it's truly unfortunate.
Anyway, I really don't know what to tell you other than I might start considering putting some distance between your daughter and this barn. It'll be hard now but it'll be even harder in 5 years when you little one feels like she's Jessie's trainer and feels responsible for her (trust me, I was there with my first lesson horse - I put so much training in on him, just to watch it continuously ruined by people who let him "bully" them and then when I offered to buy him, my offer was refused on account of the improvement my training had made in his behavior). It was so painful for me, "good" life experience but terrible for me emotionally.
It sounds like she's cuing Jessie 100% correctly for a stop. If she really "has to" keep working with this horse, I'd have her "forget" about "hurting" the horse with the bit for a few stops. She needs to "Ask, Tell, Demand" - asking = gently pulling on the reins+sitting deep, telling = pulling harder, demanding = pulling with every ounce of might she has and MAKING Jessie stop. I can't guarantee that'll work since Jessie is probably getting away with not stopping with a lot more kids than just your daughter, but in any case "Ask, Tell, Demand" is what she should always be doing with horses. Most horses, after they respect their handler (after going to the "demand" stage a few times), will only require "ask" but she needs to be willing to go to the "demand" stage. With horses, you have to seem like that old cliche of a man with nothing to loose - horses will walk over you if they think they can, you gotta make sure they believe you'll do what it takes to get the job done - no matter what. And she doesn't have to be "mean", she just has to seem like she'll go there if the horse pushes her.
Has she ever watched a herd of horses? Watching a herd of horses for a couple hours every couple of days gave me a lot of insight into how I can treat horses in a fair but confident way. There's a lot of interesting things she could learn by just sitting outside a fence, watching a group of horses. :)
I'm sorry for how long this is! I got passionate!
I hope it's not too overwhelming...