Here's an outcomes-oriented perspective:
In skiing, every action we take has a delayed outcome. The way we balance at the top of the turn dictates how effective the bottom of our turn is. Similarly, one turn's exit determines how well we can enter the next turn. You've probably felt before that when a run starts out well it's easy to keep it going, but when one turn goes badly, it becomes hard to recover. To get you the most improvement in your skiing, I'd recommend improving the top of the turn, since it will improve all the remainder of the turn.
Presently you enter the top of each turn with a push away from the ski and down the hill. You're anticipating where you want to be and trying to make yourself move there. By pushing away from the ski, you get an abrupt pressure on the edge. Pressure is good since you need it to redirect your ski. However, once you've moved away from the ski, you lose that pressure, and now you have no way to keep steering that ski around; your performance suffers and most of your turn becomes a smear. You've literally put yourself inside of the turn for its entirety because of how you started the turn.
So what can you do instead to get the skis to turn, and to manage your speed? The turning part is easy. Your friend, gravity, wants to pull you down the hill. Once your skis are tipped on an edge, without any pushing, your skis will turn around the arc. Any pushing or pulling of that outside ski just interferes with the turning effort since it disrupts that pressure. (I know, this will be too fast - we'll get to speed control later. For now, stay with me on how we can get effective turning to happen.)
There are two good ways you can help the ski design and gravity to turn you:
First, you can help put the skis onto the edge, not by pushing in, but by balancing over the outside ski. If you pretend you're a marionette and there are strings attached at your hips and shoulders, think about "lifting" or "lightening" the inside shoulder strings so that you balance on that outside ski. It will feel like you're balancing up the hill at the start of each turn. You absolutely can only develop your balance on the greens and then bring it into steeper pitches.
Some exercises to balance early on the ski include thousand steps; if you can do thousand steps from the top of the turn, you're balancing early on the outside edge
Hopping through the entire turn; same idea - if you move your body down the hill, you won't successfully hop
Skating into your turns: John Gillies explains it here:
Second, you can increase your edge angle and encourage stronger natural balance by creating some separation. When you have a bit of natural lead change (not forced) and your hips face more outside the turn, it's much easier to allow your body to angulate or bend over the outside leg. When your shoulders and hips point in the same direction of your tips, it's very hard to let your upper body bend much over the outside leg. Here's my favourite instructor teaching skiers how to introduce separation into their skiing:
Both separation and angulation are required for effective edging, which provides grip. (As you improve your ability to balance on edge, you'll be able to use your joints to start aiding more in the tipping effort and to power the turn; but let's get the big balance issue sorted out first.)
The above sounds easy, but our brain gets in the way of good turns. When skiers think they need to turn and try to make turning happen, 95% start pushing themselves into the turn, twisting their legs around, or otherwise get all out of balance, ultimately wrecking their turns. What's more effective is to think about your job as balancing over the ski, and maintaining the pressure by managing the inside half of your body. If you get separation, angulation, and tipping, your skis will go on their sides, your balance will create pressure against the ski, and the skis will arc you across the hill.
So all the above is a good intro to arcing turns. But as you mentioned, you're on steeper terrain and you don't want to go that fast (which is completely reasonable!). Good skiers manage speed through the top of the turn by feathering or surfing it before you get on edge, and then carving the bottom to accelerate. Your ski will not decelerate while it's on the edge, and yet you need to use your edges to maintain speed and redirect your momentum into the new arc. So your goal is to bleed speed at the top of the turn by feathering, surfing, or drifting the top of the turn, and then gradually putting on the gas again by tipping the skis onto edge. Think of a drift turn in a car - you'll allow your skis to surf sideways a bit, then carve through the end.
But all of this must be done by setting your balance up over the new outside leg, rather than by pushing yourself into the turn. In doing so, you'll steer the top into the right shape, bleed off some speed, then arc the bottom half (or third) and carry your mass across the hill, which sets you up for an awesome new arc.