I end up with that classic kind of issue of bringing the ski around a bit too quick while getting it up on edge a bit too quick (I think the two go together), resulting in a lack of continuous edge grip and skidding the ends of the turns. The turns all start okay and then sort of go wrong partway through.
I'm getting there, slowly, with lessons
Great question. I may have a bit of a vague goal when you put it like that! I'd say I want to be able to do skiing which is visually recognisable as carving. I know I'm not, because sometimes when I'm skiing and I'm thinking WOW I THINK I AM CARVING??, I get a friend to take a video of me and discover that in fact, no - not even close!! I think the turns should be rounder, cleaner, less snow spray at the ends which would mean that I'm not skidding the skis halfway through the turn. The skis should tip on edge and the tails should be following the tips, and the skis should be following a curve to the outside of my body as I go through a turn (so there is a little bit of angulation and inclination).
Worthy goals, good description, rounder, cleaner should be in your immediate sights. Forget about what you look like. Concentrate on what the skis do on the snow. The fact that you are skidding halfway in the turn indicates... what? Are you in a hurry to make the turn happen? Why?
I think it starts farther back. I think your turn initiation leads you away from carving and into skidding. Spend a lot of time on an uncrowded slope very easy for you. Round out the turns, feel the edges bite. Round out the beginning as well as the middle and the end. Be patient. Follow the edges, do not flatten and swish. At the end of the turn,
do not "get the turn started,
do not "bring the skis around." Leave that behind for now. As you finish the turn, come over your skis and roll evenly, smoothly, both skis rolling in unison onto the new edges. That's all. Feel the new edge bite the snow. Stay on that edge. Be an ice skate, not a toboggan. As the turn starts to happen, let the pressure build. Then slowly release, flatten the ski still riding the edge, and roll onto the other edges for the next turn. After a while, play with higher and lower edge angles to modify the turn radius. On a gentle slope, you might only make turns 20 degrees each way. It's important to find the edges such that you feel zero skid, zero slippage. Worry less about turn radius. If you need to, stop and start over. If you can't keep from going too fast, find a gentler slope.
As you go a little faster, as you pressure the skis more and bend them, as you tip the skis more, the radius decreases. Forces will build exponentially. So it gets harder in a hurry.
It's good that you're taking lessons. Ask your instructor to teach you "railroad tracks"