In high velocity, linked turns where the developed Centripetal Force is by far and away the most dominant, the gravitational terms Weight and Topple are not in play. When teaching a beginner or instituting a drill at low velocities, the reverse is true. This is why some drills fail to solve a problem.
"Weight" being a function of mass and gravitational force, it's probably more useful to think in terms of mass than in terms of weight, and then to think in terms of the total force experienced by the skier. Mass is one component determining centripetal force. Added to centripetal force is the force that results from gravity--technically, I think, what the skier experiences is the resulting ground reaction force. So the total force pushing up against the feet of a skier in a turn is the combination of centripetal force and the ground reaction force that gravity creates. To put what geepers has said in other words, at any isolated moment in the turn, a blindfolded skier wouldn't be able to tell what fraction of that total force resulted from gravity.
I'm sure everyone participating in this forum is a good enough skier to, without even thinking about it, use the energy of the old turn to move dynamically into the new turn. Beginners aren't typically good at this. If a skier maintains balance fully resisting the forces of the old turn until the last moment, and then releases all at once, the skier's momentum will tend to carry the skier's center of mass in a straight line. The only thing that will start the skier's center of mass moving toward the center of the new turn will be gravity--which as you've noted becomes a relatively weak component of the forces acting on the skier, as velocity increases. In that case, if the skier wants to shorten the radius of the new turn, the skier's only options will be to push off the hill, stem, make a pivoting move, or do some combination of these. These are the movement patterns one typically sees in beginners.
On the other hand, if the skier begins releasing the center of mass from balance against the forces of the old turn,
while the skis continue traveling the arc of the old turn, then the center of mass will be able to move more rapidly into the new turn.
This pattern of an early progressive initiation of release has been given many names--toppling, "climbing the wall," "the infinity move," "using the force." I believe they really all amount to the same thing.