On the track where I race the first corner at the end of the straight is taken at about 130MPH. It takes a LOT of effort to get the bike to turn in- even my little RS250.
Being aware of how ones control inputs are affecting the machine are really necessary to understanding what one is doing. You will find that you actively and consciously push or pull the bars left to go right and vice versa once you have done it a bit and thought about it. You can get a bike to lean over RIGHT NOW by forcefully counter steering, provided your speeds are reasonable. Get going really fast and you can work up quite a sweat.