What are some tips to making sharp turns in NFS Carbon?

I notice that you didn't mention about gears anywhere in your question. In order to turn efficiently without losing speed, you'll need to manipulate the gears. MT is always faster than AT, given similar skill. That's because AT shifts the gear too early, which reduces your top speed. You'll also want to have drift types for such corners.

There's no one-trick for all types of sharp turns, so without showing the exact corner/curve you're trying to turn into, I can only give you my personal technique to tackling every turn for the first time. This will need to be adjusted depending on the type of car and the exact corner:

When approaching the corner, I would jam my normal brakes (not handbrake) and turn, while dropping my gear by 2. You should not need to turn too much or the car will end up in oversteer and your bumper might hit something. When you feel your vehicle stabilising in mid-turn, you can move your gear back up by one and continue from there.

One strategy I used successfully in Need For Speed is to also follow the racing line. Start at the outside portion of the turn, pass the inside portion at the middle of the turn, and finish at the outside again.

In this way, you effectively reduce the sharpness of the turn and thus can follow this path at a greater speed.