  The intra-cluster medium is prone to turbulent motion that will contribute to
the non-thermal heating of the gas, complicating the use of galaxy clusters as
cosmological probes. Indirect approaches can estimate the intensity and
structure of turbulent motions by studying the associated fluctuations in gas
density and X-ray surface brightness. In this work, we want to constrain the
gas density fluctuations at work in the CHEX-MATE sample to obtain a detailed
view of their properties in a large population of clusters. We use a
simulation-based approach to constrain the parameters of the power spectrum of
density fluctuations, assuming a Kolmogorov-like spectrum and including the
sample variance, further providing an approximate likelihood for each cluster.
This method requires clusters to be not too disturbed, as fluctuations can
originate from dynamic processes such as merging. Accordingly, we remove the
less relaxed clusters (centroid shift $w>0.02$) from our sample, resulting in a
sample of 64 clusters. We define different subsets of CHEX-MATE to determine
properties of density fluctuations as a function of dynamical state, mass and
redshift, and investigate the correlation with the presence or not of a radio
halo. We found a positive correlation between the dynamical state and density
fluctuation variance, a non-trivial behaviour with mass and no specific trend
with redshift or the presence/absence of a radio halo. The injection scale is
mostly constrained by the core region. The slope in the inertial range is
consistent with Kolmogorov theory. When interpreted as originating from
turbulent motion, the density fluctuations in $R_{500}$ yield an average Mach
number of $M_{3D}\simeq 0.4\pm 0.2$, an associated non-thermal pressure support
of $ P_{turb}/P_{tot}\simeq (9\pm 6) \%$ or a hydrostatic mass bias
$b_{turb}\simeq 0.09\pm 0.06$, in line with what is expected from the
literature.
