Yes, you are pulling yourself around with the new outside hand...swinging the right hand around to make a left turn. Bad news. At this point in your skiing, you want to do nothing with your upper body except balance.
You need to learn to ski with your feet. Right now you're kicking your heels out and bracing yourself on the downhill ski while you're leaning back toward the hill heavy on your uphill ski.
Try this on green terrain that is easy for you---preparing for a right turn, hold your self stationary, balance over the ball of your left foot, lift the tail of your right ski (tail only) off the snow an inch or two, keep the ski tips even. You'll find that a stance width equal to your hip width works well. Next, aim a bit downhill, do exactly the same thing, and turn your feet to the right to make your skis turn right. Just balance with your upper body while your zipper continues to face downhill (never in the direction you're turning). Ski in a curve to the right until you stop. Do it again at a slightly steeper angle. And again. And again. When you run out of hill, do several to the left. You are learning these very important things--balance of the ball of the outside foot, stay very light on the inside foot, make round turns, turn with your feet, not your arms & shoulders. Continue these past the point where you're totally bored. Now, on very easy terrain, link these turns. Be sure to lighten the downhill foot at the end of one turn and the first thing at the beginning of the next turn. This is the first movement to start the turn--switch your weight from the old outside foot to the uphill foot which in a moment will become your new outside foot. As soon as you're lightened the old outside foot and balanced over the ball of your new outside foot, turn that foot under your body without cranking around your arms & shoulders. As you're ending a right turn, now balanced over your left ball of the foot, move your balance to the ball of the right foot, lift the tail-only of the left foot an inch off the snow, and turn your right foot under your body to make a left turn. Continue balancing on the ball of your right foot through the left turn. Continue this drill until you are past totally bored, then gradually increase the difficulty of the terrain. If you revert to your old habits, deduct one beer from your end-of-day allowance and return to easier terrain.
When skiing for real, we do not lift the ski but we do want to lighten it by shifting our weight.
In deeper snow, we do not want to rely on twisting the feet around to turn the skis. A more advanced movement, but not contradictory to what I listed above, is to roll your skis on edge, especially the inside ski, so the edged skis bank around the turn in the snow (not on the snow) and the skis turn you. You don't turn your skis. Also, if you aren't skiing on a firm base, you want equal weight on both feet. Cmick isn't yet ready for this stage, but when he gets balanced over his skis and no longer pulls himself around with his arm & shoulder, he will be ready.