  We study the possibility of realizing a growth rate of matter density
perturbations lower than that in General Relativity. Using the approach of the
effective field theory of modified gravity encompassing theories beyond
Horndeski, we derive the effective gravitational coupling $G_{\rm eff}$ and the
gravitational slip parameter $\eta$ for perturbations deep inside the Hubble
radius. In Horndeski theories we derive a necessary condition for achieving
weak gravity associated with tensor perturbations, but this is not a sufficient
condition due to the presence of a scalar-matter interaction that always
enhances $G_{\rm eff}$. Beyond the Horndeski domain it is possible to realize
$G_{\rm eff}$ smaller than Newton's gravitational constant $G$, while the
scalar and tensor perturbations satisfy no-ghost and stability conditions. We
present a concrete dark energy scenario with varying $c_{\rm t}$ and
numerically study the evolution of perturbations to confront the model with the
observations of redshift-space distortions and weak lensing.
