Graphene has gigantic potential in the development of advanced spintronic devices. The interfacial interactions of graphene with semiconducting transition metal dichalcogenides improve the electronic properties drastically, making it an intriguing candidate for spintronic applications. Here, we fabricated bilayer graphene encapsulated by WS2 layers to exploit the interface-induced spin-orbit interaction (SOI). We designed a dual gated device, where the SOI is tuned by gate voltages. The strength of induced SOI in the bilayer graphene is dramatically elevated, which leads to a strong weak antilocalization (WAL) effect at low temperature. The quantitative analysis of WAL demonstrates that the spin relaxation time is 10 times smaller than in bilayer graphene on conventional substrates. To support these results, we also examined Shubnikov-de Haas (SdH) oscillations, which give unambiguous evidence of the zero-field spin-splitting in our bilayer graphene. The spin-orbit coupling constants estimated by two different measurements (i.e., the WAL effect and SdH oscillations) show close values as a function of gate voltage, supporting the self-consistency of this study's experimental results. The gate modulation of the SOI in bilayer graphene encapsulated by WS2 films establishes a novel way to explore the manipulation of spin-dependent transport through an electric field.