Photons that have a helical phase front, that is, twisted photons, can carry a discrete, in principle, unlimited, but quantized amount of orbital angular momentum (OAM). Hence, twisted single photons constitute a high-dimensional quantum system with information-processing abilities beyond those of two-level single-photon qubits. To date, the generation of single photons carrying OAM has relied on a non-linear process in bulk crystals, for example, spontaneous parametric down-conversion, which limits both the efficiency and the scalability of the source. Here, we present a bright solid-state source of single photons in an OAM superposition state with a single-photon purity of g(2)(0) = 0.115(1) and a collection efficiency of 23(4)%. The mode purity of the single-photon OAM states is further examined via projection measurements. Future developments of integrated quantum photonic devices with pure OAM states as an additional degree of freedom may enable high-dimensional quantum information processing.