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We find a characterization for Fano 4-folds $X$ with Lefschetz defect$\delta_{X}=3$: besides the product of two del Pezzo surfaces, they correspondto 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 arerational. We give the list of all possible targets of such contractions.Combining our results with the classification of toric Fano $4$-folds due toBatyrev and Sato we provide explicit examples of Fano conic bundles from toric$4$-folds with $\delta_{X}=3$.

A characterization of some Fano 4-folds through conic fibrations