Non-equilibrium demography impacts coalescent genealogies leaving detectable, well-studied signatures of variation. However, similar genomic footprints are also expected under models of large reproductive skew, posing a serious problem when trying to make inference. Furthermore, current approaches consider only one of the two processes at a time, neglecting any genomic signal that could arise from their simultaneous effects, preventing the possibility of jointly inferring parameters relating to both offspring distribution and population history. Here, we develop an extended Moran model with exponential population growth, and demonstrate that the underlying ancestral process converges to a time-inhomogeneous psi-coalescent. However, by applying a non-linear change of time scale - analogous to the Kingman coalescent - we find that the ancestral process can be rescaled to its time-homogeneous analogue, allowing the process to be simulated quickly and efficiently. Furthermore, we derive analytical expressions for the expected site-frequency spectrum under the time-inhomogeneous psi-coalescent and develop an approximate-likelihood framework for the joint estimation of the coalescent and growth parameters. By means of extensive simulation, we demonstrate that both can be estimated accurately from whole-genome data. In addition, not accounting for demography can lead to serious biases in the inferred coalescent model, with broad implications for genomic studies ranging from ecology to conservation biology. Finally, we use our method to analyze sequence data from Japanese sardine populations and find evidence of high variation in individual reproductive success, but few signs of a recent demographic expansion.