Why do we feel a jerk forward when a moving bus suddenly stops?

This is explained by Newton's first law of motion, also called the law of inertia. Inertia is the tendency of a body to keep doing what it was already doing. While the bus moves forward, your whole body is moving forward at the same speed. When the driver brakes suddenly, the bus and the lower part of your body slow down because the seat and floor push on you, but the upper part of your body tends to keep moving forward at the original speed due to inertia. That is why your upper body lurches forward. The opposite happens when a bus starts suddenly: your body tends to stay at rest and you feel pushed backward. Wearing a seatbelt provides the force needed to stop you safely.