I took a run out to Squamish yesterday for lunch. I was taking it easy, in no hurry, just working on my smoothness through the corners. At some point I noticed I was stiff-arming through the turns. As a result the bike was leaned over, but I was almost upright. I started concentrating on leaning forward and dipping my shoulder and hip into the turn (I wasn't going fast enough to require hanging off). It was amazing how much more connected to the bike I felt, and how much smoother I was negotiating the turns.
Now I felt more in control and I could pick up the pace substantially while still feeling comfortable. I was better able to sight through the turn without getting distracted, or worried that I was too hot. It was a neat feeling, and it's remarkable how a slight adjustment to body position can make such a huge impact on how confidently I ride.
So to all you newbies out there: take a look at your body position when you're cornering. Make sure you are leaned with the bike, not sitting on top of it. If you find yourself making a lot of midcorner adjustments, that might be a hint to you that your body position on the bike could be improved.