Hes not bad at all he just needs work and time. To slow the jog, lower the head and stop some of the jarring before you go to artificial aides or anything try this.
A well balanced pleasure horses body works like an accordion. The back end is the power and pushes the horse along the front end collects and contains the power. When the back end is engaged and used for impulsion the gait is smoother and can not only collect better, but extend and also stop and turn much faster. Some horses tend to string out the back end and use the front end to drag themselves along. That makes for a faster, more jarring ride that is harder to get the horse to do anything quickly or smoothly.
When you ask for the jog or trot push the horse forward and help him engage his back end first. This will immediately feel faster but much more powerful. Sit up straight, keep your weight back on your seat bones and your legs under you when you ask the horse to go. Leaning forward and putting your weight on the front end is simply telling to horse to move forward and quickly, not exactly what your looking for. Once the back end is engaged it will feel like the horse is pulling his bum under himself and using it to push himself forward. Then gently and smoothly with your hands use half halts to collect him. A half halt is a smooth steady pull for only a second or two, feels like gently bumping to the horse. While you are still engaging the hind end and using the half halts the picture will look like this.
Push with the seat and legsengage the back endhalf haltjog jog jog.half halt.job jog jog.
It helps to keep a very steady calm voice and especially if the horse is excited talking and telling them easy and slow in a calm gentle voice. When you are not actively using a half halt the reins should be relaxed and smooth with no hanging on or unnecessary contact. With work the horse will drop the head, balance themselves and slow down a bit. Using the bridle only for the half halts and steering totally with your seat and legs also helps a bunch. When you get the impulsion from the back end and the collection and some relaxation from the front end praise the horse wildly. For my old horse anyways also the way to get his attention to work or turn to a new task is a gentle squeeze and a smooth half halt at the same time. To the horse this means drop the head, pay attention, relax and its time to get to work.
Im quite literally 9 months pregnant right now or else I could make you a very detailed video of exactly what Im talking about on my old horse. But I hope my explanation helps you a bit.