Now this may seem obvious but riding a bike isn't like driving a car.
: "Derrrr.....Jools tell us something we already know"
Jools: "Clear off rodent, I'm just about to"
When you steer a car, unlike countersteering, you steer in the direction the bend goes, and you hold the wheel on the appropriate amount of lock and steer all the way round the bend, only straightening up on the exit right?
:"Errrrr...right"
Well when you're on a bike, you do indeed countersteer whether you know it or not but... you only countersteer to
initiate the turn, get the bike pointing at the apex and to get it cranked over. Then you let the front wheel sort itself out and do what it needs to do to get round the bend.
So whether you push, pull or both the action is (for me anyway 'cos I push with my palms) "Push..Relax". The quicker you push, the faster the bike will crank over, the further you push the further the bike will crank over...and the effort you need to push will increase with speed.
That's why practising the CSS techniques, where the idea is to go in deep, flick the bike in using 'quick steering' and minimise the time you're cranked over is really hard (especially on the road). Why? Because the tendency when you push the bar fast, is to also push it too far and you end up having to sit the bike up again because you've steered too far and are about to run 3 feet inside the apex instead of clipping it. So it takes a lot of practise to push the bar quickly and accurately - but it's worth it.
So remember, as you start to take a corner the sequence is...Look at the apex, push...relax. The more relaxed you are the better it will all feel (oooooohhhh Matron).
Please don't do what I did when I started to learn this stuff and go "Push...and hold" (like in a car), because if you do the bike will keep cranking over until you're going down the road on yo A55...as the scars on the right hand side of my bike will witness