The only time you'd be "drafting" in a corner is if the corner is already a wide-open corner to begin with, in which case, just try and keep your tires in the same place the tires of the car in front was at a given point. For what you're experiencing, there may be several different issues causing it.
1) The car in front is able to hit the brakes harder, allowing him to drive away on corner entry.
2) The car in front has more downforce / grip, allowing him to take the corner faster outright.
3) You're hitting the brakes too early and being overly cautious -or-
3b) You're hitting the brakes too hard and too late and overdriving the corner.
If you're hard on the brakes while going through a sharp turn, then the car is very likely to understeer, meaning you have to hit the brakes harder to get the car slowed down or take a risk and lay off the brakes to let the car actually turn. You weren't very specific in where the issue happens (corner entry or exit), so here's the general rules I try to follow.
On sharp turns (think Monza 6/7 or Laguna Seca hairpin), you'll ideally want to be done braking right as you're about to start turning, so that the moment you turn the wheel, you're just getting off the brake and/or maybe getting back on the throttle slightly to help your corner speed/exit. Absolutely, though, you should be 100% off the brakes by the time you get to the middle or the sharpest part of the corner. From there, you should be using throttle control to manage your speed - more throttle if you can handle it, less if you're going too fast.
On more medium or light turns (think Suzuka East or Indianapolis NASCAR-1), you can probably get away with some brake as you're beginning the turn, but probably not much more than 50-75%, if that. You'll want to be back on the throttle asap and using that to control yourself through the turn. Do NOT be hard on the brakes on those corners - just ride them in until the car feel comfortable, then start using the throttle to manage corner exit.
By all means, braking is the hardest part of any track. It just takes some practice.