||We demonstrated a high-performance top-gated graphene field-effect transistor (FET) with excellent mechanical flexibility by implementing a copolymer gate dielectric via a solvent-free process called initiated chemical vapor deposition (iCVD). The iCVD process enables pin-hole free and conformal deposition regardless of surface chemistry of target substrate due to surface-growing mechanism based on adsorbed reactants. Accordingly, the ultra-thin polymer dielectrics can be directly deposited on top of graphene channel without any defect or pin-hole. Here, we intentionally introduced a polar functionality into a bulk dielectric layer through copolymerization. The synthesized copolymer dielectric can systematically tune the Dirac voltage (Vdirac) of graphene FETs according to the composition of the copolymer dielectric. The field effect mobility of the graphene FET with the copolymer dielectrics increased with the approach of the Vdirac to neutrality due to suppression of long-range Coulomb scattering. Furthermore, the copolymer dielectric enables the graphene device to exhibit substantially enhanced mechanical flexibility.