What makes a skier turn when weight is applied on the front-inner side of the outer ski? A downhill skier is on the slope indicated by the arrows (arrow head indicating the direction of the incline).
The skier has nearly reached the left edge and wishes to turn. The skier is heading, naturally, "north" or downhill, and the skis are oriented as illustrated in the diagram below.
The skier must put nearly all weight on the leg at the outside of the turn—the left leg in the figure.
Further, the skier must put the weight on the red parts (front-inside) of the skis.

What causes turning when weight is applied on the front-inner side of the outer ski?
In this diagram the sides of the skis are drawn as perfectly straight lines, as they used to be until the 1970s. Since skiers turned at that time just fine, but maybe not as well as with today's skis that are narrower under the feet than at the two tips, it is likely that the physics of turning can be explained based only on the straight-edge skis (with perhaps an amendment, or a sequel question if it's too complicated, to argue for the improvement with curved skis).
 A: If skis were not flexible, it would be very difficult to turn on skis.  
In skiing straight ahead with skis parallel to the direction of motion, the skier's weight pushes straight down in the middle of the skis and their natural flex results in a shape that has no left- or right- bias relative to the snow. It's like a segment of a cylindrical band, with the cylindrical axis parallel to the ground.  This is good, because the tips of the skis are always pointing slightly upwards, which helps them ride on top of the snow.
When the skis are tilted a bit sideways to the left, the flex still results in basically the same shape, but now the cylindrical axis is tilted.  The tips of the skis now point slightly to the left as well as upwards, so the ski tips experience a force to the left, causing the skis and skier to turn.
An analogous principle applies to the "trucks" (the wheel mechanism) under a skateboard: with weight on the left the front wheels toe left, while the rear wheels toe right.
A: I am not sure your theory is correct.
I have been skiing for decades and the way they taught us, which I hear is being modified with the new technology and "Carving paradigm", was as follows.
If you are trekking on an slanted path as per your diagram, keep most of you weight on the downhill leg, while the left foot is a few inches ahead and just follows along.
Then, just before the start of you turn to right reduce weight on the back of the skis by sitting into your knees and suddenly lifting up and to the right with your torso while planting the right pol in front right to act as a center for your turn circle. 
This will apply a right turn torque to both of the skis at the same time lifting most of the weight off them, giving them momentum to turn right. During the turn the weight is equally distributed to the skis.
After the end of your right turn again you put most of your weight on the left foot, downhill foot, and aim for the next left turn.
The benefit of this model is if you practice enough to be proficient in it you can take advantage of the previous turn momentum to get into the next turn effortlessly and stablish a smooth flow, weaving your way down, and will be able to ski all day not getting tired. 
The new skis are already cut on the sides as hyperbolas and for turning all you need to do is lean into the turn. The skis will do the rest of the work, carving their turn! 
A: When you put the weight as you've shown, that creates a torque on your body to move it to the right.  As you bring your body and your knees to the right, this has the effect of rolling the skis to the right.  Instead of being flat on the slope, they will start to dig in on the right edge.

In this diagram the sides of the skis are drawn as perfectly straight lines, as they used to be until the 1970s.

(Quite a bit past the '70s, I think).  But the lines are only straight in plan view.  When viewed from the side, the lines are not straight.  Your body weight causes them to sag in the center.  When combined with a roll to the right, the skis are now curving to the right and initiating your turn. 
The modern shaped ski simply enhances this effect and requires less sag to accomplish.
Front-to-back does not matter so much (at least not for the physics of the turn).  If you get your weight too far back, you will have less control, but can still find yourself turning in the same direction.
