Think of a spinning diskette which rotates around a fixed axis. What you'll find is that as you get further away from the axis as the object is spinning, the angular velocity,
\[\omega=\lim_{\Delta t \rightarrow 0} (\Delta \theta/\Delta t) = (d \theta/dt)\]\[(where: \Delta \theta \rightarrow Angular Displacement, \Delta t \rightarrow Time)\]increases with the increased displacement from the axis of rotation. Because of that, angular momentum,
\[l^\rightarrow=r^\rightarrow \times p^\rightarrow=m(r^\rightarrow \times v^\rightarrow),\]will also increase as as you get further from the axis of rotation. The outward moving vector "r" is what determines the imposed force that moving out directly from the center. This is otherwise known as the centripetal force and can be calculated as
\[F_{centripetal} = m(v^2/r)\]The further you measure from the axis of rotation the greater this force will be. The centripetal force is proportional to the square of the velocity which means that a doubling of velocity will require four times the centripetal force to keep the motion in a circle. Because the centripetal force, angular velocity and angular momentum increases and inertia,\[I=\int\limits_{}r^2dm,\]decreases as you move further away from the axis of rotation, the arc will measure greater than that of an object rotating at a slower velocity, with a slower momentum.
Thus, if a car is moving with a much higher velocity, with a much greater momentum, it will cause it to have to make a greater arc to make the appropriate turn in a given time. If it were traveling at a lesser speed, it would make a smaller arc to accommodate the turn in the same time frame.
You can feel it when you make turns around a corner in a vehicle. The force that you feel is the reactant, centrifugal force. You sort of inadvertently try to keep your body straight while turning but if you're moving at a greater speed, it's much harder to do. When you move faster going around a turn, it's as if you're rotating on the outer edge of the diskette with respect to the axis of rotation. When you move slower around a turn, it's as if you're moving closer with respect to the axis of rotation.