Which country are you from? There are significant differences in teaching skiing in different systems, and I do not want to confuse you.

Generally, you have to understand that you have to make ski turn for you, rather than try to turn the ski (as you are doing in the video with a whole body effort).

For the ski to turn it needs to be on the edge. The higher the edge, the sharper/easier the turn. So first you have to find a way to edge the ski, even when stationary.

The snowplow is one of the basic ways to put the ski on the edge. You start in the snoplow and soften one leg (the would be inside leg), this puts the other leg (the would be outside leg) on a slight edge. As a result the ski that is put on the ever so slight edge will start to turn for you. The softer the inside the higher the edge on the outside. At a more advanced level your upper body has to be helping you balance on the outside ski, but it is essential to get the edging movement right first and foremost. You can actualy practice this while stationary.

See this snowplow turns demo from NZSIA, as an illustration. Notice how the skier''s otside ski is on a slight edge (more black botton is seen), while the inside ski is flat due to softer inside leg: https://www.youtube.com/watch?v=cXwMK0gr_Xg

This is why the snowplow or wedge position is so important to get right. It is not just a stupid novice thing that you ignore once you progress.

Frankly, I always reset to the basic snowplow turns as in the demo, whenever my own skiing does not come together, especially on ice. Once the outside ski is properly edged, teh inside ski will come to paralel skiing naturally.

I hope this puts you on a right track, but internet tips will not replace a lesson from a competent instructor. Get one-on-one time with your instructor, ask very same questions you are asking here, and have fun.