Does Newton‘s second law hold true for an observer in a car as it speeds up, slows down or rounds a corner?

Newton's second law applies to any body in motion, and as
such, yes it applies to a passenger in a vehicle.  A person in a vehicle is travelling
at the same velocity that the vehicle they are travelling in is.  When the vehicle
speeds up or slows down the effective velocity of the passenger does as well.  When the
car turns the passenger is subject to the same centripetal force that the car is subject
to.

External forces applied to the car are transferred to
the passenger only if some part of the vehicle exerts that force on the passenger (like
a seat belt).  This explains why when a car comes to an abrupt stop, the passenger is
still moving forward unless some part of the vehicle no longer in motion exerts a force
on the passenger.