Large-scale exfoliation of defect-free and few-layer graphene by an inexpensive and environmentally-friendly route has been a significant challenge for a long time. Here we show that high-quality, few-layer graphene with high stability and low defect content can be obtained from natural graphite via a simple stirring process in urea/glycerol, with yields of up to 12 wt.%. We also demonstrate that this facile method can be applied to the exfoliation of other two-dimensional materials, such as molybdenum disulfide and boron nitride. The as-prepared graphene was further composited with polyvinylidene fluoride (PVDF) and the composite exhibited a low percolation threshold of 0.05 vol.%. The incorporation of low (4.5 vol.%) graphene amounts led to a significant increase in the thermal conductivity of the graphene–PVDF composites.