First thing to understand is that you don't keep countersteering all the way on the turn.
As said before, you countersteer for a split second just to lean or roll the bike to the desired side, and then you steer to the "correct" side to continue the turn.
Imagine going very fast on a car and doing a quick turn to the left: you will feel the car leaning or rolling to the right. The same happens with the motorcycle: you countersteer to the left, the bike leans to the right (where is you want it leaning) and then you make or let the handlebars turn to the right to continue the curve.
You must be aware that when you lean a standing still motorcycle or bicycle to any side, the front wheel/handlebar will turn to the same leaning side in proportion to the lean. The same happens if the bike is moving.
So, you lean your bike by countersteering and then "let" the front wheel/handlebar turn by itself to the correct side as said above.
All this is done in the background by our brain without none of our rational thinking. In fact, if we insist on using our rationale and tell our brain to do a full "correct" side turn of the handlebar, we wont be able to turn the bike (at speeds around 10mph and above) and we will only have a weaving straight line ride (and crash).
In this video http://www.youtube.com/watch?v=QSVr3bux1Q8&hd=1
go to 0:34 and see how the tires going along just by the side of the white line sudenly move to their left (countersteer) before turning to their right.
You can see the same thing, but is a little less obvious, at 0:23. Here, the bike is moving slowly towards the white line and then suddenly and quick it moves over the line (on it's left) before turning right.