  The aim of this work is to establish numerous interrelated gradient estimates
in the nonlinear nonlocal setting. First of all, we prove that weak solutions
to a class of homogeneous nonlinear nonlocal equations of possibly arbitrarily
low order have H\"{o}lder continuous gradients. Using these estimates in the
homogeneous case, we then prove sharp higher differentiability as well as
pointwise gradient potential estimates for nonlinear nonlocal equations of
order larger than one in the presence of general measure data. Our pointwise
estimates imply that the first-order regularity properties of such nonlinear
nonlocal equations coincide with the sharp ones of the fractional Laplacian.
