The edges of graphene-based systems possess unusual electronic properties, originating from the non-trivial topological structure associated with the pseudospinorial character of the electron wavefunctions. These properties, which have no analogue for electrons described by the Schrödinger equation in conventional systems, have led to the prediction of many striking phenomena, such as gate-tunable ferromagnetism and valley-selective transport1, 2, 3. In most cases, however, the predicted phenomena are not expected to survive the strong structural and chemical disorder that unavoidably affects the edges of real graphene devices. Here, we present a theoretical investigation of the intrinsic low-energy states at the edges of electrostatically gapped bilayer graphene, and find that the contribution of edge modes to the linear conductance of realistic devices remains sizable even for highly imperfect edges. This contribution may dominate over that of the bulk for sufficiently clean devices, such as those based on suspended bilayer graphene samples. Our results illustrate the robustness of those phenomena whose origin is rooted in the topology of the electronic band structure, even in the absence of specific protection mechanisms.