For simplicity, is the short answer. The Lagrangian does not *need* to be quadratic for physics. My first instinct was some kind of drag force, but general cases were worked out quite some time ago (1955):

http://prola.aps.org/abstract/PR/v99/i2/p587_1

Essentially, since in many physics problems we do not consider non-constant accelerations or time-dependent potentials (nothing like $U(x,v,t)$), $\dot{q}^2$ is associated to the kinetic energy and only the kinetic energy.

EDIT: Concrete example. $L=\frac{1}{2}m\dot{q}^2-\lambda \dot{q}^3$, potential here is $U=\lambda \dot{q}^3$. Euler-Lagrange gives us

$m\ddot{q}-3\lambda (2\dot{q}\ddot{q})=0$

So that potential gives equations of motion with constant velocity. Not exactly what we expected. So, things get strange when we want to associate Lagrangians to 'usual' phyisics problems unless we just say "quadratic in $\dot{q}$!"