  We find a characterization for Fano 4-folds $X$ with Lefschetz defect
$\delta_{X}=3$: besides the product of two del Pezzo surfaces, they correspond
to varieties admitting a conic bundle structure $f\colon X\to Y$ with
$\rho_{X}-\rho_{Y}=3$. Moreover, we observe that all of these varieties are
rational. We give the list of all possible targets of such contractions.
Combining our results with the classification of toric Fano $4$-folds due to
Batyrev and Sato we provide explicit examples of Fano conic bundles from toric
$4$-folds with $\delta_{X}=3$.
