Electron acceleration by a short high-current relativistic electron bunch (EB) in plasmas at three characteristic densities is studied by particle-in-cell simulation. It is found that if the EB is appropriately matched to the background plasma, the blowout space-charge field of the EB can accelerate the trailing bunch electrons at very high energy gain rate. This high energy gain, as well as the large-amplitude wakefield, the turbulent small-scale electron plasma waves, and the formation of large current peaks, are studied. The evolution of the EB, its blowout field, and other related parameters are shown to be self-similar.