There are a couple airplanes out there that use the winglet as the rudder, but why doesn't the winglet have a rudder on it to increase the turn of the aircraft?
The purpose of the rudder is to control the heading of the aircraft. It turns the aircraft by creating an aerodynamic moment about the vertical axis of the aircraft. The moment is product of the aerodynamic force and the arm.
The aerodynamic force is the the side force on the rudder, the arm is the distance between the rudder and the centre of gravity of the aircraft.
If the rudder would be on the winglets, the side force of the rudder would act at a short distance behind the centre of gravity, reducing the effectiveness very much.
It would be possible the have airbrakes on the winglets / wingtips and use differential braking to create the rotating moment. This is done for example on the B2 which doesn't have a vertical stabilizer. The added complexity and weight increase makes it an unattractive option for conventional aircraft.
- Vertical tails create more yawing moment per unit of drag.
- Vertical tails create less adverse rolling moment for a given yawing moment.
The rudder works by creating a side force, just like a wing creates lift. This incurs a small amount of drag (about a tenth of the side force).
Even on a swept wing their lever arm in lengthwise direction is small compared to the wingspan. If the winglets would be used as rudders, they would be most efficient as drag devices, because then their force will act on a longer lever. This has been done on flying wings (note that the SB-13 tailless glider has a maximum rudder deflection of 70° in order to create drag).
SB-13 in flight (picture source)
This makes the use of a rudder vastly more efficient for creating a yawing moment than using winglets.
The deflection of winglet rudders will also change the lift distribution on the wing, so they do not only create a yawing moment, but also a rolling moment. Vertical tails wich are high above the roll axis also add a rolling moment, but this is comparatively small to the rolling moment of a winglet with rudder on a high aspect ratio wing. In both cases, the rolling moment acts against the desired roll direction, so it is preferable again to create the yawing moment with the vertical tail.
It is possible, but just not that useful.
In order to maximize the moment incuded by the rudder (or keep the rudder as small as possible to provide this moment), you want to place the rudder as far (in longitudinal sense) from the center of gravity. This increases the moment induced, and thus the effectiveness of the rudder. In a normal airliner, you'll have a tube with wings in the middle, and thus the most aft part to place the rudder is at the end of the tube. On the aircraft in your picture, the most aft part is the wing tip, so it makes sense to put the rudder there.
If you place the rudder at the end, this will mean there will be substantial forces induced at the tip of the wing if you use the rudder. This means that you will have a substantial bending moment added to the wing, making it much heavier. This is further discussed in Winglet vs span extension
The existing answers miss what seems to be a fairly obvious answer: the winglets are an afterthought, added to existing, proven designs (e.g. 737) to improve fuel economy. Putting rudders on them would require running control cables, perhaps strengthening the wing structure, &c, rather than just bolting on new tips.
You could perhaps design a new airliner from scratch to do this, but again, why abandon a proven basic design unless there are large gains to be had?