We give a geometric interpretation of Bar-Natan's universal invariant for the
class of tangles in the 3-ball with four ends: we associate with such 4-ended
tangles $T$ multicurves $\widetilde{\operatorname{BN}}(T)$, that is,
collections of immersed curves with local systems in the 4-punctured sphere.
These multicurves are tangle invariants up to homotopy of the underlying curves
and equivalence of the local systems. They satisfy a gluing theorem which
recovers the reduced Bar-Natan homology of links in terms of wrapped Lagrangian
Floer theory. Furthermore, we use $\widetilde{\operatorname{BN}}(T)$ to define
two immersed curve invariants $\widetilde{\operatorname{Kh}}(T)$ and
$\operatorname{Kh}(T)$, which satisfy similar gluing theorems that recover
reduced and unreduced Khovanov homology of links, respectively. As a first
application, we prove that Conway mutation preserves reduced Bar-Natan homology
over the field with two elements and Rasmussen's $s$-invariant over any field.
As a second application, we give a geometric interpretation of Rozansky's
categorification of the two-stranded Jones-Wenzl projector. This allows us to
define a module structure on reduced Bar-Natan and Khovanov homologies of
infinitely twisted knots, generalizing a result by Benheddi.