Aside from actually turning the handlebars at slow parking lot speeds 100% of motorcycle turning is done, consciously or subconsciously, by counter steering.
Bingo! And if you want some technical data I did some extensive research on this 10 years ago. See
http://koczarski.com/Motorcycles/MotoGeometry/Info.htm.
You can do two things to carry more speed through a turn.
1. Lowering the riders weight would allow you to either decrease the lean angle for a given speed and turn radius, increase the speed for a constant lean angle and turn radius, or tighten the corner for a given speed and lean angle.
2. Shifting you weight to the inside of the turn also gives you the same options. You will increase you peg clearance if you
remain at the same speed and turn radius. You can either increase your speed or tighten the turn if you keep the same lean
angle.
If you simplify things and make the assumption that the height of the mass of the bike + the riders is constant there is one lean angle for any given turn radius and speed. If you sit straight up on the bike you and the bike will be at that lean angle. If you push the bike down but sit up like the guy in the picture below, an imaginary line through the composite center of mass of the bike and the rider is still the same for the given turn radius and speed.
As other have said, you can ride through the turn sitting straight up, tip the bike down and counter lean or hang off the bike and have the bike more upright.
If you are too hot for a given turn you have several options. You can slow down. You can take the turn wider and hope you don't run off the road You can lean more to the inside sitting straight up relative to the bike and hope you don't run out of clearance. Or if you're hanging off the bike with it tip more to vertical, counter steer a bit more and pull the bike down towards you and tighten your turn radius. My opinion is you are in a far better position just pulling the bike down to you.
The original posters question had to do with not being prepared and caught off guard when the turn comes up. If you're not already hanging and shifting your body it's going to be hard to adjust that situation much once you're into the turn. You can drop the inside elbow and shift your body pretty easily and give yourself a little more margin compared with just leaning the whole system, you plus the bike, in a straight up orientation.
Watch a MotoGP rider approaching a turn. They shift their butt and start hanging before the turn, then they counter steer, fall into the turn and adjust the bikes angle as their knee slides across the tarmac. They are basically in a fixed position and they adjust the angle of the bike for the path they want to take.
We're not on a track or hanging and leaning anywhere near a MotoGP rider but the principle is the same.
The guy on the right is tipped about the same as the guy on the left but he's lowered his mass to the inside. The composite angle of him and his bike is leaned over more. He's either making a sharper turn or going fast than the guy on the left.
My 2 cents worth.....