I believe the confusion can be resolved first by realizing that directions are important in your question. And second by understanding the microscopic origin of pressure.
The first observation then concerns the fact that force is a vectorial quantity and the definition of pressure you give is not general enough. So the force that should be taken into account is the one done perpendicularly to the surface $A$. Alternatively you can consider an area $A$ as a vector too, then the general definition would include $\cos\theta$, where $\theta$ would be the angle between the force and area vector.
For the second point, intuitively, think that pressure is produced by particle collisions which exert a force on a surface (by delivering momentum). Having said that, Bernoulli's equation concerns mostly pressure due to height differences and perpendicular velocities (mostly), no regard to external forces or directions. So a way to think about it is the following. Think about a flat horizontal surface (perhaps a house roof), under normal conditions particles in the air fly in all directions and collide with the roof producing a certain pressure (let's say atmospheric pressure). When the wind starts blowing strongly then most particles in the air will have a horizontal velocity so vertical collisions will be reduced, namely less particles will exert force on the roof since most of them are flying horizontally. The consequence is that the force is reduced, hence the pressure on the roof is reduced (so if the wind is hard enough to reduce the pressure to a value below the pressure within the house (ignoring how it is attached) it will get blown.) So indeed more velocity is less pressure at a macroscopic level for directions that are perpendicular.