How locally injected turbulence spreads in space is investigated with direct numerical simulations. We consider a turbulent flow in a long triply periodic box generated by a forcing that is localized in space. The forcing is such that it does not inject any mean momentum into the flow. We show that at long times a statistically stationary state is reached where the turbulent energy density in space fluctuates around a mean profile that peaks at the forcing location and decreases fast away from it. We measure this profile as a function of the distance from the forcing region for different values of the Reynolds number. It is shown that, as the Reynolds number is increased, it converges to a Reynolds-number-independent profile, implying that turbulence spreads due to self-advection and not due to molecular diffusion. In this limit, therefore, turbulence plays the simultaneous role of cascading the energy to smaller scales and transporting it to larger distances. The two effects are shown to be of the same order of magnitude. Thus a new turbulent state is reached where turbulent transport and turbulent cascade are equally important and control its properties.