We consider a non-homogeneous generalised Burgers equation:$$∂u/∂t + f (u) ∂u/∂x − \nu ∂^2u/∂x^2 = η^ω , t ∈ R, x ∈ S^1 .$$Here, $\nu$ is small and positive, $f$ is strongly convex and satisfies a growth assumption, while $η^ω$ is a space-smooth random "kicked" forcing term. For any solution $u$ of this equation, we consider the quasi-stationary regime, corresponding to $t ≥ 2$. After taking the ensemble average, we obtain upper estimates as well as time-averaged lower estimates for a class of Sobolev norms of $u$. These estimates are of the form $C \nu^{−β}$ with the same values of $β$ for bounds from above and from below. They depend on $η$ and $f$ , but do not depend on the time $t$ or the initial condition.