  We study the production of forward di-jets in proton-lead and proton-proton
collisions at the Large Hadron Collider. Such configurations, with both jets
produced in the forward direction, impose a dilute-dense asymmetry which allows
to probe the gluon density of the lead or proton target at small longitudinal
momentum fractions. Even though the jet momenta are always much bigger than the
saturation scale of the target, $Q_s$, the transverse momentum imbalance of the
di-jet system may be either also much larger than $Q_s$, or of the order $Q_s$,
implying that the small-$x$ QCD dynamics involved is either linear or
non-linear, respectively. The small-$x$ improved TMD factorization framework
deals with both situation in the same formalism. In the latter case, which
corresponds to nearly back-to-back jets, we find that saturation effects induce
a significant suppression of the forward di-jet azimuthal correlations in
proton-lead versus proton-proton collisions.
