Semiconducting transition metal dichalcogenides (TMDs) are promising for flexible high-specific-power photovoltaics due to their ultrahigh optical absorption coefficients, desirable band gaps and self-passivated surfaces. However, challenges such as Fermi-level pinning at the metal contact–TMD interface and the inapplicability of traditional doping schemes have prevented most TMD solar cells from exceeding 2% power conversion efficiency (PCE). In addition, fabrication on flexible substrates tends to contaminate or damage TMD interfaces, further reducing performance. Here, we address these fundamental issues by employing: (1) transparent graphene contacts to mitigate Fermi-level pinning, (2) MoOx capping for doping, passivation and anti-reflection, and (3) a clean, non-damaging direct transfer method to realize devices on lightweight flexible polyimide substrates. These lead to record PCE of 5.1% and record specific power of 4.4 W g−1 for flexible TMD (WSe2) solar cells, the latter on par with prevailing thin-film solar technologies cadmium telluride, copper indium gallium selenide, amorphous silicon and III-Vs. We further project that TMD solar cells could achieve specific power up to 46 W g−1, creating unprecedented opportunities in a broad range of industries from aerospace to wearable and implantable electronics.