Aircraft create lift by accelerating air downwards. The lift created is the equal and opposite reaction that is predicted by Newton's third law.
The wings acelerate the air downwards by creating pressure differences.
Bernouli's theory is one method of explaining how the wings create the pressure differences.
It is worth noting that the lift is not created by the downward velocity of the air, but by the downward acceleration given to it. This might appear to be splitting hairs, but giving the air a large downward velocity means giving it lots of kinetic energy. This loss of energy from the aircraft means drag. If however a very large mass of air is given a very small downward acceleration, it will be given less kinetic energy so there will be less drag.