Why can a motorcycle rider at normal riding speed start a right turn by briefly pushing the right grip forward instead of steering it like a car?

Answer: A brief countersteer

A car-like front turn — Almost, but this imports car intuition into a single-track vehicle. At normal riding speed, the motorcycle first needs a lean setup before it can follow the right-hand curve. Pointing the front wheel directly into the turn skips that setup and can create the wrong initial response.

A brief countersteer ✓ — Correct. Pressing the right grip briefly steers the front wheel a little left, shifting the tire contact path out from under the bike and letting the motorcycle lean right. The surprise is that countersteering is not the whole turn; it is the quick setup that creates the lean needed for the real right-hand curve.

A deeper body lean — Leaning the body helps tune the corner, but it is not the first reliable road-speed command. The motorcycle needs the wheels to move under the bike so the machine can lean. That is why riders are taught the small handlebar press rather than trying to throw their torso at the curve.