Microswimmers exhibit an intriguing, highly-dynamic collective motion with large-scale swirling and streaming patterns, denoted as active turbulence -- reminiscent of classical high-Reynolds-number hydrodynamic turbulence. Various experimental, numerical, and theoretical approaches have been applied to elucidate similarities and differences to inertial hydrodynamic and active turbulence. These studies reveal a wide spectrum of possible structural and dynamical behaviors of active mesoscale systems, not necessarily consistent with the predictions of the Kolmogorov-Kraichnan theory of turbulence. We use squirmers embedded in a mesoscale fluid, modeled by the multiparticle collision dynamics (MPC) approach, to explore the collective behavior of bacteria-type microswimmers. Our model includes the active hydrodynamic stress generated by propulsion, and a rotlet dipole characteristic for flagellated bacteria. We find emergent clusters, activity-induced phase separation, and swarming, depending on density, active stress, and the rotlet dipole strength. The analysis of the squirmer dynamics in the swarming phase yields Kolomogorov-Kraichnan-type hydrodynamic turbulence and energy spectra for sufficiently high concentrations and strong rotlet dipoles. This emphasizes the paramount importance of the hydrodynamic flow field for swarming and bacterial turbulence.