Turbulent magnetic fields abound in nature, pervading astrophysical, solar, terrestrial and laboratory plasmas. Understanding the ubiquity of magnetic turbulence and its role in the universe is an outstanding scientific challenge. Here, we report on the transition of magnetic turbulence from an initially electron-driven regime to one dominated by ion-magnetization in a laboratory plasma produced by an intense, table-top laser. Our observations at the magnetized ion scale of the saturated turbulent spectrum bear a striking resemblance with spacecraft measurements of the solar wind magnetic-field spectrum, including the emergence of a spectral kink. Despite originating from diverse energy injection sources (namely, electrons in the laboratory experiment and ion free-energy sources in the solar wind), the turbulent spectra exhibit remarkable parallels. This demonstrates the independence of turbulent spectral properties from the driving source of the turbulence and highlights the potential of small-scale, table-top laboratory experiments for investigating turbulence in astrophysical environments.