  Many-body localization is characterized by a slow logarithmic growth of the
entanglement entropy after a global quantum quench while the local memory of an
initial density imbalance remains at infinite time. We investigate how much the
proximity of a many-body localized phase can influence the dynamics in the
delocalized ergodic regime where thermalization is expected. Using an exact
Krylov space technique, the out-of-equilibrium dynamics of the random-field
Heisenberg chain is studied up to $L=28$ sites, starting from an initially
unentangled high-energy product state. Within most of the delocalized phase, we
find a sub-ballistic entanglement growth $S(t)\propto t^{1/z}$ with a
disorder-dependent exponent $z\ge1$, in contrast with the pure ballistic growth
$z=1$ of clean systems. At the same time, anomalous relaxation is also observed
for the spin imbalance ${\cal{I}}(t)\propto t^{-\zeta}$ with a continuously
varying disorder-dependent exponent $\zeta$, vanishing at the transition. This
provides a clear experimental signature for detecting this non-conventional
regime.
