How much effect does the Bernoulli effect have on lift? I understand that the Bernoulli effect is a flawed explanation for the cause of lift, and does not cause much at all, but how much? 
Is there any experimental data on the force caused by the Bernoulli effect? Maybe implicitly through data of the pressure difference between the top and underside of an aeroplane's wings. After that, I assume I could (crudely approximating the pressure to be acting perpendicularly to the flight direction) use $\Delta P A$ to work out the net force on the plane.  
Perhaps there is another way to quantitatively analyse the extent to which the Bernoulli effect causes lift.
Edit: see this short cartoon (content similar to Mike Dunlavey's answer).
 A: Sometimes you will see  statements like 'some of the lift is caused by Bernoulli's principle and some of it is caused by Newton's laws", but this is the wrong way to think about it. The fact is that 100% of the lift can be explained by Newton's laws and 100% can be explained by Bernoulli's equation. Both approaches explain 100% of the lift.
The problem is that popular explanations using Bernoulli's principle usually make a dog's breakfast out of things, so I understand why you think it's a "flawed" explanation  - usually it is.  However, applied correctly Bernoulli's equation can be used to predict 100% of the lift.
To grossly oversimplify things, classical aerodynamics calculates lift something like this:


*

*write down the differential equations expressing conservation of momentum, conservation of energy, and conservation of mass.

*solve those differential equations and apply boundary conditions to get a solution

*the solution is a vector field that represents the speed and direction of the airflow at each point in space 

*Forget about the direction and just use the magnitude of the vector - this is the speed

*Now that you know the speed everywhere, use Bernoulli's equation to substitute pressure for speed

*integrate the pressure over the airfoil surface to get the lift.


Done correctly, this number is 100% of the lift. So, Bernoulli's principle is responsible for 100% of the  lift. The thing is, almost all of the physics is in step one - the rest is a bunch of hairy math and then a calculational trick at the end to use Bernoulli's equation to turn speed into pressure. This trick (mis)leads some into thinking Bernoulli's equation somehow "explains" the physics, but without the context of the rest of steps it makes little sense. 
If you look at the overall context you'll see that Bernoulli's equation is a small and relatively unimportant part of the overall theory. But correctly applied, it predicts 100% of the lift.
A: There's no problem with the Bernoulli effect, only with the way it's understood and explained.
It's usually explained with mistakes, like the need for asymmetrical airfoil and equal flow time above and below, and without mentioning the need to deflect the direction of airflow.
Here's the best light-math explanation I've seen.
Also study this section that directly answers your question.
EDIT: It is easy to find wrong pictures like this:

as opposed to a correct one like this (from the link above):

So the answer to your question is: All of the lift depends on the Bernoulli principle, because speed and pressure are in trade-off, but the physics need to be correctly understood.
A: I take it as self-evident that lift can be generated by a wing without any difference whatsoever in the shape of the top or bottom of the wing. After all, balsa wood gliders (or ones that are rubber band powered) fly just fine with completely flat wings. If you look at such a plane, you will see the wing is set at an angle relative to the longitudinal axis of the fuselage: the wing is tilted upwards at the front relative the rear. In normal flight, the relative wind strikes the bottom of the wing and pushes it upward (Newton's law). Curvature of the upper surface of the wind improves the lift due to the Bernoulli effect. (But from what Mike Dunlavey says above this is not necessary for the Bernoulli effect to operate.) I have understood that the other factor producing lift in a wing is "ram effect" caused by the wind striking the bottom of the wing due to the slight positive angle of attack the wing has in level (cruise) flight. Now planes can fly inverted, but since the wing's fixed angle of attack and the so-called Bernoulli effect (or whatever it is) now operate in conjunction with gravity to pull the aircraft toward the ground, the pilot must fly with an exaggerated nose-up attitude to maintain level flight. This allows the wind to strike the upper surface of the wing (remember, the plane is inverted) with sufficient force to compensate for these factors. This additional ram effect keeps the plane flying when inverted. I think M. Dunlavey might subsume what I am calling ram effect under his interpretation of a correct explanation using the Bernoulli effect. If so, I have no problem with it. But I find the concept of ram effect more accessible, and it has good lineage, going back to the Wright brother's concept of center of pressure of the wing. 
A: I cannot say exactly how much the Bernoulli Effect contributes to lift but it is not much. Cambered wings and barn doors fly inverted. Air speed increase is the same as the increased distance ratio. not enough to produce much lift. The wing tries to generate a ring vortex. This gets thwarted because the roll up around the trailing edge, called the "starting vortex" is shed before takeoff. This leaves, in plan view, a horseshoe shaped vortex consisting of the "bound vortex" and the two tip vortices. Stretch your mind here! The "Bound Vortex" is not convection, it is diffusion. It is a molecular knock on effect and travels at sonic speed around the wing forward underneath and stream direction on top and rolls up around the tips. Some air follows the diffusion around the tips and that is convection. The pressure above a wing is not reduced because of accelerated air, The pressure is reduced first by the bound vortex which then accelerates the air. The wing is a pump. Air from 18 feet above a Cessna 172 is accelerated down at 5 tons/second in normal flight. Thats how Newton is involved.. For every action.....  Before responding check a text book on aerodynamics. See also www.newfluidtechnology.com.au "The Coanda Effect and Lift".  
A: I see so many people trying to get this and still making some mistakes. 
 Mike Dunlavey and Paul Townsend have a lot correct, but still miss something very important that puts this all completely to rest.
First:
Bernoulli’s Principle and Equation are two completely different things  Please stop conflating them.
You need to understand that Bernoulli’s Principle/Theorem/Effect does not cause anything, nor explain the science.  
Bernoulli’s Principle only points out the inverse relationship between pressure and speed -- in a special (and very limited) situation AND, it must be observed in a special way (I’ll explain that below ***).   It says nothing about what causes what.   In addition, the “increase in speed” mentioned in the classical wording is an acceleration and that is the most important take-away from this.  
Euler, in the mid 1700s followed up on Bernoulli’s work and determined that Pressure Gradients Accelerate Fluid and derived Bernoulli’s Equation. <-- this is critical!
Then:
Bernoulli’s Equation can be used to calculate the lift value.  So when Paul says: “100% can be explained by Bernoulli's equation.” This is correct, but incomplete and misleading.  Integrating The Equation over the area of the wing does give the lift force, but you still don't know what causes the pressure in the first place.
Bernoulli’s Equation does not explain the science of the cause, it only gives the upward force value, but not why it happens in the first place.
When Paul also says: “100% of the lift can be explained by Newton's laws” same thing, you get the lift value from the down-wash momentum transfer, but it is also incomplete.
So what’s the real deal?
When a wing moves through the air, this relative motion directly causes pressure differences at the surfaces. 
It is that simple.!.
A.  The pressure increase under the wing is caused by the simple fact that the wing is advancing on the air, pushing on it.  That makes an increase of pressure. This is Adak47‘s “RAM effect” put in better terms.
B. The pressure reduction above the wing is a bit more complex to describe, but is due to the simple fact that the air and wing are ‘trying’ to move away from each other (the opposite effect of “ram”).
Then, and most important:
These very same pressures do two things at the same time:
1. They are the pressure difference that pushes up on the area of the wing.
2. They also cause all of the accelerations of air that we see around the wing: Ahead, below, above, behind and around the tips.
In other words, the pressure difference and the downwash/deflected air are both TWO Parts of the  Whole story, not two different ways to look at it.
They ARE, however, two ways to calculate the force.
that is what everyone is missing.
Second:
Stop focusing only on the wing shape.  What is critical is the air-flow pattern around the wing and it is NOT the same as the wing’s shape. The fact is that ALL wings produce very similar flow patterns.  This goes for symmetric, asymmetric, inverted and flat.
Then, one more thing to blow your minds before I provide a reference explaining this all in detail.
If you stand on a hill and watch a wing go by, the fastest moving (highest speed) air is under the wing!! Yep!, This is measured and actual data on a wing generating lift. This blows the doors off this “fast air” nonsense, because it shows that acceleration of air is what is key, as I mentioned above.
What is going on is that the lower air is pushed forward pretty fast in the direction of flight.  However, the upper air is accelerated rearward, really yanked rearward, with high acceleration.!  The reduced pressure is part of the Euler Pressure Gradient that causes this acceleration.  It is actually traveling rearward at slightly lower speed than the lower air is traveling forward…
This is all called out more carefully here:
https://www.quora.com/q/kyuoyckftflurrpq/Understanding-Bernoulli-s-Principle-Correctly 
https://www.quora.com/q/rxesywwbdscllwpn/Understanding-Lift-Correctly 
Regards, Steve
P.S. The bound vortex (circulation) also doesn’t cause anything, but it is something we can observe because it happens to be an effect of the way the air is accelerated near the wing. 
In linear fluid motion.
***  When the pressure upstream is higher than downstream, the net force is forward, thus speeding up (positive acceleration) the air.  This is called  “proverse” pressure Gradient.
This is the ONLY case that the classical words called “Bernoulli’s Principle” describes.
The opposite is also true. 
When the pressure downstream is higher than upstream, the net force is rearward, thus slowing (negative acceleration) the air.  This is called an “adverse” pressure gradient.
In addition,
When the Pressure Gradient is across the stream, the flow is curved.  This is what curves (or “turns”) the flow both below and above a wing.
I’ve studied this and discussed this with two noted experts actually in the field and who have practiced aerodynamic design, Boeing’s Doug McLean and Embry-Riddle’s Charles Eastlake.
A: I'm sorry but the accepted and most voted answer ("All of the lift depends on the Bernoulli principle") here can't be right according to:

*

*The accepted and most voted answer to What really allows airplanes to fly? (planes fly because they push enough air downwards and receive an upwards lift thanks to Newton's third law)

And this video  explains why: the different speeds of airflows above  and below a wind is a consequence (not the cause) of a previous  difference in pressures.
A way to understand this, is to think in a plane flying through perfectly still air. The wing accelerates quiet air (i.e. speed = 0) to moving air with two speeds: speed_above > speed_below.
This means that the differences in speed can't be the cause of the differences in pressure: the air was at speed 0.
Finally, if it was not the speed of air: what caused the differences in pression? Maybe beacuse of the air being deflected down?
The fact that the Bernoulli equation can be used to correctly calculate the force of lift does not implies the Bernoulli effect is responsible for the lift.
If the force due to pushing air down is the actual cause of the lift (lift is the same amount but in the opposite direction) you can still calculate the orginal force and thus the lift by measuring a direct consequence of such downwash: the differences of pressure.
A: When you are in the passenger side of a car cruising at a speed above 30mph, as 50mph to about 70mph provides a better sample, just roll down the window and extend your hand outside with the palm down, now tilt it slightly back and feel the 'ram air'
trying to push your hand in an upward motion. This is all you need to know to understand how a wing flies as the upward angle of attack relative to the still air
(tilt back of your hand) is about 1 to about 4 degrees before it starts to get nearer to the aerodynamic stall point while providing most of the lift necessary for sustained level flight. If Bernoulli effect provided much life there would be no need for all airplanes to 'rotate' on takeoff when reaching V1.  Read the book Stick and Rudder to get more into the explanations as the author also dispels the Bernoulli effect as being much of a force in providing wing lift.
A: My above is CORRECT science.  I talked with MIT's Mark Drela, Teacher/author Charles Eastlake and teacher/author John D. Anderson & others.  Bernoulli explains/causes NOTHING! It only describes the observed pressure/speed inverse relation in some situations, but gives NO cause for anything. Speed DOES NOT cause any pressure change. In short - Acceleration is caused by Pressure Gradient (Euler).   Read my posting above.  Here's a short video better explaining why pressure is lowered along a convex curved flow (above a wing) It is correct: physics: https://www.youtube.com/watch?v=3MSqbnbKDmM
