You can also get your weight to the inside of a turn for a bicycle by leaning while riding straight.
Every time this comes up there is a big debate with a bunch of people saying counter steering is required. Please, go out and try it. Drive perfectly straight along a painted line. Lean left, turn left. Lean right, turn right. There is no requirement for counter steering.
I think it's somewhat speed-dependent. I ride motorcycles and bikes. On a motorcycle at high speed it's impossible (in my experience) to take a turn without counter steering. You sometimes have to push really hard on the handle on the side where you want to go.
Also, leaning is very similar to counter steering; counter steering makes the bike "fall" on the side where you want to go.
Interestingly, you can also steer a horse without doing anything on the reins; the horse will usually go where you put your weight; I think it's because it needs to compensate for the weight differential; or maybe it takes it as a hint about which way you're looking. In any case it works.
Every time this comes up there is a big debate with a bunch of people saying counter steering is required. Please, go out and try it. Drive perfectly straight along a painted line. Lean left, turn left. Lean right, turn right. There is no requirement for counter steering.