To turn on a dime is to change direction almost instantly. Your neighbor's huge SUV may be powerful, but it can't turn on a dime like your tiny Smart car.
When you literally have a very small turning radius, you can turn on a dime, whether you're on foot or in a speed boat. This phrase is also commonly used figuratively, to mean instantly change focus or switch abruptly to something new. Your older sister might talk pleasantly about her day for a while, only to turn on a dime and suddenly demand to know if you borrowed her bike without asking.