The PHENIX collaboration presents first measurements of low-momentum (0.4<p_{T}<3 GeV/c) direct-photon yields from Au+Au collisions at sqrt[s_{NN}]=39 and 62.4 GeV. For both beam energies the direct-photon yields are substantially enhanced with respect to expectations from prompt processes, similar to the yields observed in Au+Au collisions at sqrt[s_{NN}]=200. Analyzing the photon yield as a function of the experimental observable dN_{ch}/dη reveals that the low-momentum (>1 GeV/c) direct-photon yield dN_{γ}^{dir}/dη is a smooth function of dN_{ch}/dη and can be well described as proportional to (dN_{ch}/dη)^{α} with α≈1.25. This scaling behavior holds for a wide range of beam energies at the Relativistic Heavy Ion Collider and the Large Hadron Collider, for centrality selected samples, as well as for different A+A collision systems. At a given beam energy, the scaling also holds for high p_{T} (>5 GeV/c), but when results from different collision energies are compared, an additional sqrt[s_{NN}]-dependent multiplicative factor is needed to describe the integrated-direct-photon yield.