Mass production of high-quality graphene sheets is essential for their practical application in electronics, optoelectronics, composite materials, and energy-storage devices. Here we report a prompt electrochemical exfoliation of graphene sheets into aqueous solutions of different inorganic salts ((NH ₄)₂SO₄, Na₂SO₄, K ₂SO₄, etc.). Exfoliation in these electrolytes leads to graphene with a high yield (>85%, ≥3 layers), large lateral size (up to 44 μm), low oxidation degree (a C/O ratio of 17.2), and a remarkable hole mobility of 310 cm² V^-1 s^-1. Further, highly conductive graphene films (11 Ω sq^-1) are readily fabricated on an A4-size paper by applying brush painting of a concentrated graphene ink (10 mg mL^-1, in N,N′-dimethylformamide). All-solid-state flexible supercapacitors manufactured on the basis of such graphene films deliver a high area capacitance of 11.3 mF cm^-2 and an excellent rate capability of 5000 mV s^-1. The described electrochemical exfoliation shows great promise for the industrial-scale synthesis of high-quality graphene for numerous advanced applications.