Why is velocity inversely related to pressure in a flow? I've seen the equations that give this relationship, and I understand the math and have seen it worked out in problems. But I don't have a qualitative, conceptual grasp on the relationship.
Is the pressure that which is exerted by a small element of the flow, or exerted on a small element? Maybe it is the pressure exerted on or by the boundary?
And once pressure is defined, why is it related to velocity? Why does something push less if you speed it up? I would have guessed a faster flow pushes harder.
 A: Be careful with directions. Usually the pressure although not a vector, can be understood as $F/A$, where $F$ is the magnitude of the force. However this one does have a direction associated to it. For the Bernoulli equation one is normally interested in the horizontal speed of the fluid and the pressure due to changes in height. With the picture I described in mind, particles with a horizontal velocity, hitting a vertical wall lead to $P$ proportional to $v$. While a horizontal surface under a fluid with horizontal speed, indeed experiences less pressure above it, you can think of less particles having the chance to hit (transfer momentum) to the surface since the are mostly traveling horizontally.
A: You say "if you speed it up".
Who speeds it up?
The only thing that can speed it up is a pressure difference.
(Let's do it horizontally, so we can ignore gravity.)
That's why less pressure means higher velocity, and vice-versa.
A: The "pressure is inversely related to velocity" formula is applied when fluid is in dynamic motion and $P =f/a$ as area is inversely to velocity. Instead, in $P = fv$ pressure is directly proportional to velocity and it is applied only on static fluid
$P〆v$ in static fluid
$P〆1/v$ in dynamic fluid
