This relation between a wing's curvature and a pressure difference on both sides is often part of the 'equal transit time' explanation; air on the curved side has to traverse a longer distance in the same amount of time, therefore goes faster, which leads to a lower pressure. This explanation is very common and completely wrong.
In normal flight, pitching the nose up causes the aircraft to climb because the wings meet the air at a steeper angle; the lift increases. It makes sense that rotating the wings in the opposite direction decreases lift. In fact, point the nose down far enough and the wings will produce no lift at all. Beyond that, the generated lift becomes negative and the wings will start to pull the aircraft down.
During our hypothetical manoeuvre, our attitude has varied by about 10°. That's not exactly flying upside down yet, the curved side of the wings were on top the entire time. Whether or not the lift was pointing up as well, depended on the angle at which the wings meet the air, the angle of attack.
The same is true for inverted flight. If we find ourselves at an attitude where the wings are pulling us down, we raise the nose. At first, the downward lift will disappear and at higher angles of attack, start pointing up and grow larger. At sufficient airspeeds and angles of attack, we have enough lift to maintain altitude upside down.
So why do wings need to be curved at all? They don't. Flat wings also provide lift at non-zero angles of attack and are perfectly usable, but not very efficient. Properly shaped airfoils create more lift and less drag. To find out why, consult a more accurate explanation of how planes really fly.