  In this paper we consider $N \times N$ real generalized Wigner matrices whose
entries are only assumed to have finite $(2 + \varepsilon)$-th moment for some
fixed, but arbitrarily small, $\varepsilon > 0$. We show that the Stieltjes
transforms $m_N (z)$ of these matrices satisfy a weak local semicircle law on
the nearly smallest possible scale, when $\eta = \Im (z)$ is almost of order
$N^{-1}$. As a consequence, we establish bulk universality for local spectral
statistics of these matrices at fixed energy levels, both in terms of
eigenvalue gap distributions and correlation functions, meaning that these
statistics converge to those of the Gaussian Orthogonal Ensemble (GOE) in the
large $N$ limit.
