How fast do you go on 90 degree turns?

I don't know if I am right or wrong, but I feel like at low speed when you turn, you turn the handlebar in the direction you want to go, but at higher speeds you push the handlebar to counter steer and make the bike lean and therefore turn where you want to go. When making a 90 degree turn at a light - say you are cruising on a road at 40mph, change lanes into the left turn lane and the light is green, so you don't need to stop, but what speed do you slow down to? Considering the fact that the crosswalk strips at the beginning of the turn and the end are slippery, not to mention the cement part of the road for water to flow across the street when raining are slippery even when dry, how fast do you go during the turn approximately?
I used to have no question about it and would make the turns smoothly, but then I dropped the bike in a slippery area water drain area (when it was dry) making a sharp turn at low speed while leaning (the front wheel slid from under me, although I did not break, I just continued with the throttle, the wheel continued to slide out more until both I and the bike were on the ground Heartbreaking to drop a brand new bike, needless to say that made me less confident on turns and I don't lean as much, but I figure I should see how more experienced people turn. When do you start the lean, when do you come out of the lean, and how fast do you go when entering the turn (before speeding up at apex)?