It is true, the wheel is more realistic but more difficlut. I bought mine when I was about 75% through, racing the fast Pro races. I couldn't begin to keep up. I put it away until I get a chance to restart from zero, and re-learn how to drive. But I already know what to do with the controls. For you, who are just learning what the cars want and how to drive, I'd suggest you switch to the DS2 controller for a while. I like to run with the left analog stick as steering and the right analog stick as gas/brake.
I will bet, even from your description, that you are braking too late. That always seems to be the problem, and I saw a big drop in my lap times when I learned to brake earlier. You should be just getting off the brakes whn it is time to turn in for the corner. This means you can be on the gas through the turn itself. Even though the AI cars may brake later and pass you coming into the turn, if you are accelerating through and out you should pass them on exit. After you pass the apex, ease off on the steering a little and let the car edge over to the outside. Less steering at thattime will aloow you to accelerate faster.
Hope this helps, and welcome to GTPlanet!