Polymer dielectric film with a large dielectric constant, high energy density and enhanced thermal conductivity are of significance for the development of impulse capacitors. However, the fabrication of polymer dielectrics combining high energy density and thermal conductivity is still a challenge at the moment. Here we demonstrate the facile exfoliation of hexagonal boron nitride nanosheets (BNNSs) in common organic solvents under sonication with the assistance of hyperbranched polyethylene (HBPE). The noncovalent CH-π interactions between the nanosheets and HBPE ensure the dispersion of BNNSs in organic solvents with high concentrations, because of the highly branched chain structure of HBPE. Subsequently, the resultant BNNSs with a few defects are distributed uniformly in the poly(fluorovinylidene-co-hexafluoropropylene) (P(VDF-HFP)) nanocomposite films prepared via simple solution casting. The BNNS/P(VDF-HFP) nanocomposite exhibits outstanding dielectric properties, high energy density and high thermal conductivity. The dielectric constant of the 0.5 wt% nanocomposite film is 35.5 at 100 Hz with an energy density of 5.6 J cm-3 at 325 MV m-1 and a high charge-discharge efficiency of 79% due to the depression of the charge injection and chemical species ionization in a high field. Moreover, a thermal conductivity of 1.0 wt% nanocomposite film reaches 0.91 W·m-1 · K-1, which is 3.13 times higher than that of the fluoropolymer matrix. With dipole accumulation and orientation in the interfacial zone, lightweight, flexible BNNS/P(VDF-HFP) nanocomposite films with high charge-discharge performance and thermal conductivity, exhibit promising applications in relatively high-temperature electronics and energy storage devices.