There shouldn’t be any counter-steering. You should be putting in an input, holding it until you feel the chassis load up, and then basically straightening your hands out at apex and letting the chassis unwind off the corner. That’s a really basic way to put it, but that’s how you drive a modern kart.
Contemporary kart chassis have tons of grip and mechanical weight jacking. Any touch of the wheel results in a very abrupt change in weight distribution and front end geometry. Moving the wheel as little as possible is the goal. And of course, then you have to take into consideration each individual corner. Every corner has different pavement transitions, grip levels, bumps, cracks, radii, widths etc. Some corners require more aggressive input and some require less. Here’s a video of me at a SKUSA Pro Tour (national-level) race a few years ago, where you can see my hands: https://www.youtube.com/watch?v=CsokNeq_gXE
Driving quickly is about managing all your inputs and coordinating them with what you’re feeling from the tires to use all traction available at all times.
I’ve been trying to post this for a few days but my account was messed up…