While the above posts have answered the question "why ?", let me answer the question "why not ?"
In many applications we actually use a linear relationship between fluid flow and pressure drop. This too is an approximation, but a reasonable one for a certain range of flow characteristics.
The reason the square root dependence is not strictly obeyed in real fluids is because of viscous loss. The corrected expression for pressure drop is :
[tex]\Delta p = \frac{\Lambda}{D} \cdot \frac{\rho v^2}{2} \cdot L~~~~(1)[/tex]
where [itex]\Lambda[/itex] is the "friction factor". For laminar flow, the friction factor is given by
[tex]\Lambda = \frac{64}{Re} ~~~~~~~(2)[/tex]
Re is the Reynold's Number, which in the laminar regime goes like
[tex]Re = \frac{vD}{\nu} ~~~~~~~(3)[/tex]
where [itex]\nu[/itex] is the viscosity.
Plugging these into (1) gives :
[tex]\Delta p = 32 \rho \nu L v~~~~(4)[/tex]
However, for this equation to hold, we must have L/D >>1, so that the fluid attains "terminal velocity" from viscous drag. Over short lengths, the behavior will look more like the square root dependence (which is only exactly true in the ideal limit L/D -> 0 ).
In terms of the flow rate, Q = vA, this gives the more familiar (Ohm's Law like) equation
[tex]\Delta p = const \cdot Q \cdot \frac{L}{D^2}[/tex]
With turbulence, the relationship becomes more complex and the notable difference is that the dependence on the diameter takes on a higher number, typically somewhere between 3 and 4.
