Countersteering is convenient, but not required. A bicycle and rider are not a single rigid body. You can simply lean to one side and you'll have to steer in that direction to keep your bike under you -- no countersteer necessary.
I am not saying people don't countersteer, only that it isn't necessary to make a turn.
Also, bicycles aren't motorcycles where the weight ratio between rider and vehicle is swapped.
If you don't feel like clamping your handlebars so they only turn one direction, try this: coast along a straight line (and outdoor basketball court is great). Then pick a direction and just lean that way. You can absolutely keep your wheels on the line until you turn in the direction you picked, with no countersteering necessary.
This "fact" came about with a video of low skill riders who can't manipulate a bike very well, or don't know what it is they're doing when they do it.
