I have a 2004 Mazda 6 (4 cylinder). It has about 135k miles on it. I'm going to try to explain what the problem is. Every now and then when I'm driving (usually when my engine has had time to warm up) my car will start jerking and hesitating kind of like it keeps jumping from one gear to another. When I hit the brakes, it jerks quicker and harder. I usually have to turn my car completely off for it to stop and then when I cut it back on it will usually be fine. This has been happening off and on for the last year moreso these past few weeks. Does anybody have an idea of what the issue might be?

 

have you attempted to reset the ECU or see if any codes were being produced first?

 

Just a wild guess based on your description...
At first I thought it might be the MAF sensor which is extremely delicate to dirt and was going to suggest cleaning or replacing it. Even though mounted after the air filter in a "closed" environment, dirt will build up on it over time.
Then when you discussed it happening under braking also....
Here's what I "think" it might be: Find the ABS sensors mounted on the suspension of all 4 wheels and see where they point to. Follow that "line" until you locate the tripper, usually a grooved or toothed section on the axle or brake rotor. See if there is dirt, rust or general crud in there and clean it out w/cleaner, a wire brush (NOT on the sensor itself), or whatever it takes. Also GENTLY clean out any dirt in or around the sensor.
In short, I think your ABS system is receiving a false signal and is working overtime.