This is why the game starts every one out in a FF car, so you can learn the dynamics of different drive train layouts.
FF cars tend to be understeery when pushed especially in corners but they are the most controllable.
FR are the main car build and are a usually built to understeer at the limit but many powerful FR cars tend to oversteer when pushed.
AWD cars can have either FF characteristics or FR/MR characteristics so you can either get an understeering car at the limits or one that has snap oversteer.
MR cars are mainly a layout used for race motors, main reason is because it places the heaviest part of the car near the center which aids in direction changes which is the greatest weapon when racing a car transitioning between movements. This layout is sharp and is like balancing on a knife edge so you have to balance throttle and steering in corners as you can brake later than any other drive train layout, then as you release the brake you can go on the throttle lightly to keep the wheels planted and as your car straightens up you can increase more throttle. Trail braking is key for controlling an MR car in corners.
RR like the big boy Ruf's this is bit like MR but even more notorious with oversteer, but like the MR cars this one requires delicate throttle control even on slow corners because steering inputs can end in horrific crashes. Once you master their nuances these cars are monster weapons because of late and deep braking into corners allowing you to get back on throttle earlier. Trailbraking works in this layout just as well.
GT2 guide book spoke of trailbraking and they also stressed that racing through corners a driver is never coasting they are always doing something, either braking or accelerating. You guys need to be patient and learn not to on/off drive MR cars, they are not as stable as FR cars especially in corners. Lift off oversteer happens when you ease up off the throttle whilst in a corner, as I said a driver is doing braking or accelerating in a corner almost never coasting. Learn to use the throttle and brakes well, learn to be gentle with steering inputs and throttle and you will see just how joyous it is to drive cars. As in real life different drive train layouts require different techniques in order to achieve that car's full potential.
Don't compare GT5 times to GT6 as GT5 had a more static tire simulation than GT6 does. In fact in GT6 I can drive any car on the stock tires with ease and countering oversteer is a simple thing once you understand what you are doing wrong and you can drill into your mind how to approach corners depending on what layout you car has and it's characteristics. If you car is oversteering or becoming unstable in corners what are you doing that is causing it? Check your throttle inputs and steering inputs, you drive the car the car doesn't drive you. If you go with the cars' dynamics and understand it you'll drive faster instead of fighting it to make it drive like some other car you drove. It's a car not a human, you cannot force it to change by being overbearing, understand it and accept it for what it is.
GT5 tire grip was insane compared to GT6, in GT6 you have a far more progressive grip loss change, it's freaking excellent. I am having so much fun that it reminds me of GT2 so much. Also stick to comfort soft and sport hard tires in GT6 for nearly every car in the game, I drive my NSX '91 on those and it's a joy to drive even when tuned to 500pp it's just too good. In GT5 there would be no way I could handle the car this well, especially with the on/off grip loss in GT5 simulation, street tires don't lose grip like that it's always progressive. GT6 finally recreates that and it's beautiful, PD finest physics/tire simulation with suspension dynamics to date. I can't wait until they really get some power to really push the boundaries of physics in a game for the masses.