There are a few things that will cause a car to jerk either to the side or while shifting gears. The average auto owner can troubleshoot these problems. The repairs will require a professional.
A damaged tire will cause a car to pull or jerk to a side when driving at medium speeds. A tire damaged by a piece of metal or glass will cause the car to run out of alignment. The best way to find this is to examine the tire visually and physically. Look for damage. If you do not see any, run your hand over the entire tire surface front and back. You should be able to feel damage.
Brake Shoe Locking
This is caused by a worn brake shoe or a lack of brake fluid. If the shoe is worn and the pads are worn, it will grip the rotor and cause the care to jerk to the right or left. Jack up the car, and remove the tire. Examine the brake pads for wear. If they are worn, replace immediately. Examine the rotor, which is what the pads grasp when stopping. If it is heavily grooved or worn thin, the pads will grab and stick on the rotor.
If the car is jerking while shifting gears, there might be two problems. First, the transmission fluid might be low. Drive the car for a few minutes to heat the fluid. Place it in neutral on a flat surface, and check the fluid level. This will be on a dipstick toward the rear of the engine. If it is low, add the recommended fluid to the proper level. If the fluid is not low and it is still jerking, the problem could be more serious and require a professional diagnostic test to examine the transmission.