ABSTRACT Accreting black holes (BHs) launch relativistic collimated jets, across many decades in luminosity and mass, suggesting the jet launching mechanism is universal, robust, and scale-free. Theoretical models and general relativistic magnetohydrodynamic (GRMHD) simulations indicate that the key jet-making ingredient is large-scale poloidal magnetic flux. However, its origin is uncertain, and it is unknown if it can be generated in situ or dragged inward from the ambient medium. Here, we use the GPU-accelerated GRMHD code h-amr to study global 3D BH accretion at unusually high resolutions more typical of local shearing box simulations. We demonstrate that turbulence in a radially extended accretion disc can generate large-scale poloidal magnetic flux in situ, even when starting from a purely toroidal magnetic field. The flux accumulates around the BH till it becomes dynamically important, leads to a magnetically arrested disc (MAD), and launches relativistic jets that are more powerful than the accretion flow. The jet power exceeds that of previous GRMHD toroidal field simulations by a factor of 10 000. The jets do not show significant kink or pinch instabilities, accelerate to γ ∼ 10 over three decades in distance, and follow a collimation profile similar to the observed M87 jet.