  We derive normal approximation bounds in the Kolmogorov distance for sums of
discrete multiple integrals and $U$-statistics made of independent Bernoulli
random variables. Such bounds are applied to normal approximation for the
renormalized subgraphs counts in the Erd{\H o}s-R\'enyi random graph. This
approach completely solves a long-standing conjecture in the general setting of
arbitrary graph counting, while recovering and improving recent results derived
for triangles as well as results using the Wasserstein distance.
