  We study the renormalization group flow of the average action of the
stochastic Navier--Stokes equation with power-law forcing. Using Galilean
invariance we introduce a non-perturbative approximation adapted to the zero
frequency sector of the theory in the parametric range of the H\"older exponent
$4-2\,\varepsilon$ of the forcing where real-space local interactions are
relevant. In any spatial dimension $d$, we observe the convergence of the
resulting renormalization group flow to a unique fixed point which yields a
kinetic energy spectrum scaling in agreement with canonical dimension analysis.
Kolmogorov's -5/3 law is, thus, recovered for $\varepsilon=2$ as also predicted
by perturbative renormalization. At variance with the perturbative prediction,
the -5/3 law emerges in the presence of a \emph{saturation} in the
$\varepsilon$-dependence of the scaling dimension of the eddy diffusivity at
$\varepsilon=3/2$ when, according to perturbative renormalization, the velocity
field becomes infra-red relevant.
