You don't counter-steer during the turn. You counter-steer to start the turn. Once the bike is leaned into the turn it will keep turning as long as the speed stays the same. If the speed increases the bike will begin to stand back up. The centripetal force pushes the bikes mass away from the direction of travel. If the bike slows down it will continue to fall closer to the ground until it maintains a speed, or hits the ground. The front wheel will turn into the turn a bit after the bike leans into the corner. The pictures below show this very well. Kudos the the member of the second photo. My apologies that I forgot your screen name.
Body position has more to do with applying a riders weight to help set the suspension up for the corner, and transferring the center of gravity lower on the bike by putting pressure on the pegs. The effective CG will drop about a foot on the bike making it more stable through the corner. If you are doing this correctly you will feel it in your thighs very quickly. After a 20 minute track session my thighs were burning. At the end of the day, and six track sessions, my legs felt like jelly. The next morning I felt like I had just run for 10 or 15 miles.
Having the knee on the ground, in my opinion, has more to do with looking cool than actual riding. If you are leaning the bike so much you can't avoid dragging a knee you better be on a race track. You can hang off a bike and never touch a knee down in the corners, even if you have your bike leaned enough to grind hard parts. MotoGP riders lean their bikes so much that their body parts are in the way. Knees dragging in that case are purely from the rider using it as a gauge for how much more lean angle they have left before they run out of traction. Racers also use rear sets that are high enough to keep their feet from dragging on the ground even when the engine casing is just millimeters from scraping.
Jellyrug is correct about the car tires. At 8x the contact patch most cars have about a 2:1 advantage over a motorcycle for traction. Also, car tires have harder sidewalls and reinforced shoulders. That helps apply more pressure to the road as the car leans away from the direction of the turn. Anti-sway bars reduce the body roll and that also puts more pressure on the shoulder of the outside tires. Keeping the tire tread surfaces parallel helps to reduce loosing grip surfaces during the corner.
This is from my own experience and opinions. I might not have all the details correct, but that it the information I have gleaned from several years of riding, attending a performance riding school, and listening to experienced racers in the pit areas between track sessions.