What you're basically doing is sideslip turning. When you're sideslipping, you're applying the brakes, so you're not really carving.
The issue is that you're going one direction, and you whip the board another direction. It will sideslip until you ease up and let the edge take you in a new direction.
To carve, the goal is to get you're board at an angle and let the edge cut an arc through the snow. You will feel your board cut and sweep in an arc across the snow. It will not feel like grinding. You have to be patient with your board. Don't swing the board around, get on the edge, get your balance, and let the board turn you. If you try to rush the turn, you'll sideslip.
I frankly think you can carve on greens, it's just hard to get the speed you need for hard carving and hard turns. But greens is a good place to start, because you don't feel the need to apply the brakes and skid. Then you can take it to blues and blacks and try harder carving.