The only time you steer left to go left is if you're going so slowly that the only reason the bike doesn't fall over is because of your near-superhuman balance. That is to say, when you're practically at a standstill. Anywhere approaching walking pace or above, if you try to turn left by steering left, you'll fall off the right hand side before you make any appreciable turn.
That's the issue; how do you lean left to start with? If the human-bicycle system has its center of gravity over the wheel line, shifting your weight relative to the bike doesn't move the center of gravity - you lean one way, the bike leans the other, and you end up in the same overall position. The only way to initiate a lean to the left is to countersteer to the right.
You are. I proved this to myself once by applying pressure to the handlebars using only the palm of my hands on the backside of each handle. Even at very slow speeds, I could not for the life of me turn left by applying pressure only to the right bar (to turn the front wheel to the left). No amount of leaning helped. To lean the bike, I needed to create some external force to push it over. Leaning my own body to the left just made the bike lean to the right.
I've seen the same claim from a lot of people who seemed to have thought about it a lot: that you always countersteer on a two-wheeled vehicle, but you only do so consciously if you're riding a very fast vehicle or you want to turn extremely quickly. (A deliberate countersteer is taught in some bicycle safety classes as an "emergency quick turn".)
This is counterintuitive to me as a regular cyclist, but I realize that most of my cycling skills are completely unconscious, so I don't have a lot of confidence in my ability to describe exactly what I'm doing on my bike.
That's what I used to think until I started taking motorbike lessons.
At first I was having a hard time turning, because this whole counter-steering thing didn't come naturally to me and I would try to turn the handlebars to the left if I wanted to go left, but the bike would go the other way witch was pretty scary.
I then internalised this and the next time I rode a bike I tried this out, and indeed, this is how it works. If I turned the handlebars left at any considerable speed, the bike would lean to the right and start turning right, with the handlebars going right. It would go a slight bit left at the very beginning, but then it would go all the way right.
When I think about it, at slow speeds (say below 25 km/h - 15 mph) I only turn the handlebars and lean the bike. However, I can lean the bike at those speeds, so no counter-steering is required. I first lean the bike and then turn the bars.
However, it's very difficult to have it lean any useful amount at speed. And this is where counter-steering is useful.
Granted, this is way more noticeable on a motorbike that on a bicycle. Maybe the weight of the wheels and of the whole bike has something to do with it. If I lean a 10 kg bike but stay upright on it, the centre of gravity doesn't move much. However, on a 300 kg motorcycle, it moves quite a bit more.
The only time you steer left to go left is if you're going so slowly that the only reason the bike doesn't fall over is because of your near-superhuman balance. That is to say, when you're practically at a standstill. Anywhere approaching walking pace or above, if you try to turn left by steering left, you'll fall off the right hand side before you make any appreciable turn.