the first thing to do if you want to make a car go faster safely is to check the suspention and the wheel alignment. also check all the rubber brushings and pieces where various suspenting compontents attach. if the rubber mounts of an anti roll bar for example are worn the car will feel much more "loose" and unstable especially during fast manouvres.
here's an intresting article on ride and handling with a little science on how humans precieve this as well
Automobile Ride, Handling, and Suspension