Why do you lurch forward in a bus that suddenly slows? why do you lurch backward when it picks up speed? what law applies here?

The law applied here is Newton's first law, also known as, law of inertia.

This law states that: A body will retain its state of rest or motion unless acted upon by an external force.

If you are moving and the bus suddenly stops, your body will lurch forward trying to retain its state of motion until it comes to rest and changes its state by the external force acted on it.

If you are at rest and the bus suddenly moves, your body will lurch backwards trying to retain its state of rest and opposing the force of motion until it is forced to change its state by this force.