Finishing or completing a turn means heading across the hill between turns. Many people have never explored what a fully finished turn feels like (wheee!), what such turns can do for you when you link them down the hill (high quality speed control), nor what it takes to make it happen in linked turns (that illusive crossing of the body over the skis).
I once watched an incredible junior racer come down the hill on slalom skis below me (I was on the chair). He had a personal coach wearing a USST jacket following him. This kid made the most beautiful, round, fully-C-shaped turns that I've ever seen. I see junior racers all the time at my mountain, but they don't complete their turns because they are focusing on going fast. This kid's focus must have been on getting the highest edge angles he could while making fully completed turns. He booted out once but did not fall; I could see the small divot in his movement just at it happened. What a skier.
For recreational skiers, it's not so much finishing the turn that's the problem. It's starting the next one afterwards. Turn initiation when heading across the hill can feel more challenging on even slightly challenging pitches. Even at slow speeds it can be daunting.
There are many ways of describing how to initiate a turn from an across-the-hill path:
-- the skier must allow the paths of the feet and the upper body to "cross" each other
-- the skier must get "upside down" at the top of the new turn
-- the skier's CoM must cross the BoS
-- the skier must move downhill of the skis.
I like the last description for simplicity.
These descriptions reveal why so many skiers don't finish their turns; they can't start the new turn if they fully finish the old turn. They have not yet worked on moving downhill of their skis to start the new turn. It can be scary, and self-preservation gets in the way of giving it a try. Learning to do this is a big milestone in the progression to higher skiing skills.
I find that focusing on where the feet go between the turns (instead of where the upper body goes) makes this initiation easier. I don't hear instructors talking about teaching this foot-focus very often. The thing to do is bring your feet back up under you AND turn the skis to point almost uphill as you do this. If you keep the skis turning until they almost point uphill, you'll be traveling across the hill at turn's end and get your C-shaped, completed turn. If you bring them back up under you as you do this, instead of having them way downhill of you, then your new turn will start automatically. It's quite the eye-opener when this first happens.
Why does this work? Because as those feet come back up under you, they will move just enough uphill of your shoulders to put you in that desired situation where "you" will be on the other side of your skis, where your CoM will be downhill of your BoS. The skis will then tip to new edges on their own, and you'll be in the new turn before you know it without having done anything to start it. Hold onto your socks!
This is an exhilarating feeling at speed.