  We derive global gradient estimates for $W^{1,p}_0(\Omega)$-weak solutions to
quasilinear elliptic equations of the form $$
\mathrm{div\,}\mathbf{a}(x,u,Du)=\mathrm{div\,}(|F|^{p-2}F) $$ over
$n$-dimensional Reifenberg flat domains. The nonlinear term of the elliptic
differential operator is supposed to be small-BMO with respect to $x$ and
H\"older continuous in $u.$ In the case when $p\geq n,$ we allow only
continuous nonlinearity in $u.$
  Our result highly improves the known regularity results available in the
literature. In fact, we are able not only to weaken the regularity requirement
on the nonlinearity in $u$ from Lipschitz continuity to H\"older one, but we
also find a very lower level of geometric assumptions on the boundary of the
domain to ensure global character of the obtained gradient estimates.
