We consider a generalised Burgers equation \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2}=0,\ t \geq 0,\ x \in S^1, where $f$ is strongly convex and $\nu$ is small and positive. Under mild assumptions on the initial condition, we obtain sharp estimates for small-scale quantities. In particular, upper and lower bounds only differ by a multiplicative constant. The quantities which we estimate are the dissipation length scale and averages for structure functions and the energy spectrum for solutions $u$, which characterise the "Burgulence". Our proof uses a quantitative version of arguments contained in Aurell, Frisch, Lutsko & Vergassola 1992. Our estimates remain true for the inviscid Burgers equation.