You don't "need to improve quite a bit". That steep terrain and heavy snow would represent a challenge to 90% of the skiing public. So let's call it an opportunity to improve, with the recognition that the farther up you get on the skill ladder, the harder it is to get from one rung to the next and the more choices you have as to which rung to step on. The bottom line here is that you are successfully navigating difficult terrain and that's a lot to be proud of.
Before I get to specific observations, the compression on this video was pretty ugly. It was hard to see the turns in detail. So, caveat emptor on the analysis.
In heavier snow, many people use an up move to help get turns started. In steeper terrain, many people are "heavy" on the bottom parts of their turns for speed control purposes. These moves are visible in this clip. In the left turn where you are across from the camera, you can see the skis in a wedge position and the upper body bent at the waist and rotating into the new turn to help it. This is a common move when we finish turns too far across the hill for speed control, we're not fully "grounded" and the snow is resisting our attempts to turn the skis. When we are in the air, upper body movements are more effective than lower body movements to get the body rotated. But in this snow, you're not fully airborne either. There is enough contact to make lower body movements effective, but not enough to do it after "landing" and sticking your skis deep into the snow (note the highest snow plumes coming off the skis at the "bottom" of the turns). The tactical solution is to narrow your turn shape a little, pick up more speed down the fall line and "plane" higher within the snowpack. Although narrower, your turns will become more rounded, smoother and more efficient. However, it's unlikely that you will be able to "just do this". You probably don't "own" the turn initiation moves needed to make these turns possible. These are movements that are more easily developed on flatter groomed trails and I'd rather see your skiing in that environment before giving you a specific recommendation for how to get there. In the mean time, it's easiest to just try a faster, narrower line in these conditions. Enjoy! It looks like a ton of fun.