The answer to this is long. The majority of traction issues, whether with TC on or off are caused by user input. That's the reality.
A quick note on acceleration issues in GT, overall: If you're allowed to, turn the rear differential acceleration setting way down. This is for RWD and rear-drive biased AWD cars only. For some reason, starting in GT6, PD set the acceleration on diffs at like 40. It's way too high for ANY rear-biased car. This high level of lock will cause your outside rear wheel to start spinning before, or at the same time as your inside rear on corner exit, and cause oversteer. I don't know why they did this. I guess maybe the drifters like it, but this ain't drifting. I recommend setting rear diff accel at 10, then tuning +1 tick at a time if the inside wheel is spinning until it stops, or -1 if you're still getting snap oversteer at corner exit, until that stops. I'll put my next paycheck down that resolves 50% of your corner exit issue right there. Now, moving on.
First, if you're not using R2 for the accelerator, don't bother. Some have luck using the right joystick for accel + braking, but I've always found it a little too hard to finely modulate throttle with it as the controller moves around while you steer/while it rumbles. So, assuming you're using the right trigger to accelerate, here's my advice:
It sounds like you're giving too much throttle input, regardless of your TC setting. Keep in mind that TC in game and in the real world works by cutting power when wheel slip is detected. If you just mash the accelerator coming out of a corner, TC is going to slow your engine speed down in order to avoid wheelspin. Not only do you lose power due to the lower engine speed, but you lose further acceleration even after TC stops intervening b/c you're RPM/gear combination will be sub-optimal. Without TC on, you would just spin the wheels and go nowhere fast (or oversteer into the wall). With TC, you're still going nowhere, but the reason is different (as stated, your engine speed has been severely reduced). Either way the result is the same: slow corner exit. The key for both approaches is to have good throttle modulation.
You are inherently at a little bit of a disadvantage at this being a pad user. The trigger has far less physical travel than a pedal does--maybe half an inch of depression compared to several inches for a pedal. Because the trigger has to provide the same range of throttle opening with a lot less travel, it's way more sensitive than a pedal. The good news is that, barring some neuromuscular issue, you should have more fine control in your index or middle finger than you do in your ankle. Practice not only being smooth with the trigger throttle, but being millimeter-precise with your inputs. Half a millimeter even. On race cars or street cars with flywheel tuning, even half a millimeter is a big difference in throttle opening and RPM. Practice feeding more throttle as your car starts to grip without just mashing it. In the same vein, practice easing off when wheelspin begins without completely snapping off throttle. It'll take a while, but you'll get a great feel for it. For a real challenge, do this on the least grippy tire that makes sense for the car you're in. On race cars, use racing hards. On powerful street cars, practice with sports hards. When you master traction with these, you'll be golden on grippier tires.
Practice the above with TC off. You will never know what the car is truly doing if you've got the computer running things. When you've got it down, go ahead and turn the TC on. Start with the lowest setting, of course. Because you learned to modulate the throttle well without TC, you'll now be a lot better at exiting corners without activating it too much, while still being able to lean on it a little if you make a small mistake or as your tires start to wear off in longer races. Turn it up as needed in especially difficult weather conditions.
Important: don't ever feel bad about using traction control. I've been playing these game since I was a teenager, and there's always been this stigma against using TC. I admit I have been both a victim and perpetrator of said stigma (hence how I got so good at driving w/ a controller without it, haha!). But the reality is, the overwhelming majority of race cars in this game use traction control in their real-life counterparts. So even the pros...the guys who get paid and get rich off driving these cars use TC. Unless it's explicitly banned from a race or league, use it! Just know that it will be an enemy and not a friend until you learn good throttle modulation for yourself. It's a cane, not a wheelchair.
