What happpens to a car moving at a constant speed as it turns a corner?

The car undergoes a change in velocity. It accelerates (and does so at a constant rate - if the rate of turn is constant) in the direction of the turn. The speed is constant, but the direction the car is going changes at each instant in time because it is turning. And because it is changing direction, that means its velocity is changing (because velocity is speed plus a direction vector).