"its a natural movement. Another example would be a plane sways while turning."
Yes, but for a complete different reason than a dancer does.
An airplane sways (banks) so that the lift produced by its wings is no longer simply up, but now directed up and towards the inside of the turn. This pushes the plane towards the center of the turn, making the flight path curve.
A dancer on the other hand sways because the lower body moves faster than the upper. This doesn't have much to do with turn - you can dance quite a bit of sway on a non-turning promenade for example.
Wen dancers do turn, they do so by first rotating their body, but the direction of travel is not yet changed. Then when their swing cycle reaches its highest and slowest point, they take their next, lowering step in the new direction towards which their body was previously rotated. To a pilot what a dance does is a skid turn, and pilots have an instrument to avoid doing this. But a dancer is not an airplane - we are supported by feet acting vertically against the floor, rather than wings which provide support in whatever direction they are banked to.
What is a coordinated turn for an airiplane - nose always pointed into the movement - is a mistake for a dancer. What is a skid turn for an airplane - pointing to the new direction while still moving in the old, otherwise known as skidding - is correct for a dancer. Different mechanics, different techniques.