This paper presents experimental scalings of the electrons and positrons produced by intense laser-target interactions at relativistic laser intensities (1018–1020 W cm−2). The data were acquired from three short-pulse laser facilities with laser energies ranging from 80 to 1500 J. We found a non-linear (≈EL2) scaling of positron yield [Chen et al., Phys. Rev. Lett. 114, 215001 (2015)] and a linear scaling of electron yield with the laser energy. These scalings are explained by theoretical and numerical analyses. Positron acceleration by the target sheath field is confirmed by the positron energy spectrum, which has a pronounced peak at energies near the sheath potential, as determined by the observed maximum energies of accelerated protons. The parameters of laser-produced electron-positron jets are summarized together with the theoretical energy scaling. The measured energy-squared scaling of relativistic electron-positron jets indicates the possibility to create an astrophysically relevant experimental platform with such jets using multi-kilojoule high intensity lasers currently under construction.