Context. The theory of jet emitting disks (JEDs) provides a mathematical framework for a self-consistent treatment of steady-state accretion and ejection. A large-scale vertical magnetic field threads the accretion disk where magnetic turbulence occurs in a strongly magnetized plasma. A fraction of mass leaves the disk and feeds the two laminar super-Alfvénic jets. In previous treatments of JEDs, the disk turbulence has been considered to provide only anomalous transport coefficients, namely magnetic diffusivities and viscosity. However, 3D numerical experiments show that turbulent magnetic pressure also sets in.Aims. We analyze how this turbulent magnetic pressure modifies the classical picture of JEDs and their parameter space.Methods. We included this additional pressure term using a prescription that is consistent with the latest 3D global (and local) simulations. We then solved the complete system of self-similar magnetohydrodynamic (MHD) equations, accounting for all dynamical terms. The magnetic surfaces are assumed to be isothermal, limiting the validity of our results to cold outflows. We explored the effects of the disk thickness and the level of magnetic diffusivities on the JED response to turbulent magnetic pressure.Results. The disk becomes puffier and less electrically conductive, causing radial and toroidal electric currents to flow at the disk surface. Field lines within the disk become straighter, with their bending and shearing occurring mainly at the surface. Accretion remains supersonic, but becomes faster at the disk surface. Large values of both turbulent pressure and magnetic diffusivities allow powerful jets to be driven, and their combined effects have a constructive influence. Nevertheless, cold outflows do not seem to be able to reproduce mass-loss rates as large as those observed in numerical simulations.Conclusions. Our results are a major upgrade of the JED theory, allowing a direct comparison with full 3D global numerical simulations. We argue that JEDs provide a state-of-the-art mathematical description of the disk configurations observed in numerical simulations, commonly referred to as magnetically arrested disks (MADs). However, further efforts from both theoretical and numerical perspectives are needed to firmly establish this point.