You are talking about the kind of vehicle I grew up and learned all my driving skills on. A 1956 Ford, pedal to the metal and wound out and sweating oil, could make maybe 115-120 mph, and anything except straight-line under those circumstances was akin to driving on a thin skim of ice. And once the tires lost adhesion to the road, well, there were no air bags, and only a simple lap belt to keep you in place.
If you were wearing one.
Oh yeah, sliding on those bench seats.
the thing is though police cars usually were souped up and beefed up- sway bars, stabilizers, rugged suspension, and overhauled engines- so they were safer and more road sticking power than the average vehicles at higher speeds-
And the dash and most everything else inside was steel.