Lots of good advice already in this thread. Lots of people I've ridden with don't realize this is a skill and something they need to get better at, so you're already ahead of the game.
This is something you can learn quickly, just do some solo practice. To bolster some of the points above:
1. Drop and weight the outside pedal. Keep your center of gravity low through a corner.
2. Lean the bike, not your body. Learn "bike/body separation". Keep your body slightly more upright than the bike.
3. Brake before a corner and accelerate exiting the corner. DO NOT brake in a corner, instead hit the corner at the right speed to take you through it.
4. Look ahead of you to where you want to go, don't focus on the road right in front of your wheels. Keep your head up.
5. If possible, consider taking up mountain biking. Singletrack will teach you how to corner.
6. Learn to ride the "racing line": the fastest, widest radiused line that clips the apex.
7. After you've practiced and mastered all of the above, consider trying different tires. Some tires just corner better.