I do drive like you with smaller cars...what I found out, it get outclass by bigger cars easily especially in corners and straights (unles you are using Turbo).

Then, when you start to drive with bigger car, you will find it is no way a sports car...but typical family car...

The typical cars you see on the road are not made for spirited driving...unless it has been modified extensively.

My advice, change shocks/springs and use extreme tires...but for your rims, extreme tires are hard to get. Once you start to modify more you will find out that your cars limits easily reach...and its not enough..with better suspension setup you can really rev the engine...then become accelerated wear and tear...