I don't want to put you off from the game, but I'm afraid most of your solution is going to be a change in driving style. It's a big switch from GT4, but once you get used to it you'll be able to go back and forth without much trouble.
Here are some tips I can give you (I'm going to try to cover all the bases, so please don't think I'm patronizing you):
- Try to brake earlier. Unlike GT4, Enthusia simulates brake lock, at least in some situations. If you fly into a corner and turn-in while braking, your rears will lock -- you might as well have pulled the handbrake. Feathering the brakes helps too, but is pretty hard to do on the DS2.
- Be careful with the throttle. Enthusia's simulation of wheelspin, and the rear-end "stepping out" under power (with RWD and some AWD cars), are both more accurate than in GT4. In the middle of a corner with a powerful-enough RWD car, it can only take a moment on-throttle to enter a slide. If you're using the DS2, I recommend using the X button instead of the stick, which has a poor analog range.
- Pay attention to weight transfer. It matters more in Enthusia than it does in GT4. Especially since lift-off (off-throttle) oversteer is
actually simulated (unlike in GT4 -- sensing a trend yet?). GT4 lets you go willy-nilly with the steering, brakes, and throttle -- it'll mostly just understeer. Enthusia requires more of a deft touch, lest you find yourself flying backwards into a wall.
- Try not to downshift while cornering. Enthusia simulates "shift-lock" oversteer, as the Japanese call it. And you guessed it -- GT4 doesn't. Downshifting mid-corner not only "jolts" the driveline and can break traction, it can also lead to off-throttle oversteer (as above).
- Practice and get a feel for the steering response (with a DS2) or force feedback (with a DFP/G25). That's pretty much the only way to learn to avoid overcorrection.
- Use a newer car. I'm not joking -- for whatever reason, Enthusia seems to give older cars "older" tires with less grip. Consequently, '60s cars are great for big, long, showy drifts.
For most people this is the biggest problem, because they went straight for the Enthusia Life mode and picked something old as a starter car (I don't really remember what there was to choose from).
Aside from all of that, there's a tuning menu, albeit a simplistic one. All you have to do is watch the bars above the settings to see what you're doing, if you didn't already know that stuff. Just try setting your car towards understeer and/or a slower cornering response.