Since you apparently did not read the link I posted, I will paste the passage for you.
The problem isn’t that the tires offer better grip, it’s the cars can’t handle it. All that extra grip puts stress on both classic cars’ suspension and chassis, something neither were built to handle. This can cause them to fail or break more readily since bias-ply tires would normally handle the extra stress. In fact, if you put a radial tire on an old car, the extra stress could cause the wheel to crack or fail completely.
That’s not to say there aren’t some workarounds if you want a smoother ride on a classic car. If you’re truly determined to put radial tires on a classic car you’ll need to swap out the old suspension for one that’s modern and able to handle the extra traction. You may lose a “classic” element from the car, but you’ll have a smoother and safer ride as a trade-off.
Did you catch that? So, as is the case in real life, you'll have to put new suspension on your car to use those tires. Because, again, as evidenced by what I shared, if you try that for real, you'll break the car. Since the suspension in the game does not break for simply driving, you get your bouncing.
So, as has been the case through this thread, the issue is USER ERROR