We report on measurements of quantized conductance in gate-defined quantum point contacts in bilayer graphene that allow the observation of subband splittings due to spin-orbit coupling. The size of this splitting can be tuned from 40 to 80 $mu$eV by the displacement field. We assign this gate-tunable subband-splitting to a gap induced by spin-orbit coupling of Kane-Mele type, enhanced by proximity effects due to the substrate. We show that this spin-orbit coupling gives rise to a complex pattern in low perpendicular magnetic fields, increasing the Zeeman splitting in one valley and suppressing it in the other one. In addition, we observe the existence of a spin-polarized channel of 6 e$^2$/h at high in-plane magnetic field and of signatures of interaction effects at the crossings of spin-split subbands of opposite spins at finite magnetic field.
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