We demonstrate the formation of a high-angular-momentum turbulent state in an exciton-polariton quantum fluid with TE-TM Spin-Orbit Coupling (SOC). The transfer of particles from quasi-resonantly cw pumped σ+ component to σ− component is accompanied with the generation of a turbulent gas of quantum vortices by inhomogeneities. We show that this system is unstable with respect to the formation of bogolons at a finite wave vector, controlled by the laser detuning. This instability can be triggered by an inhomogeneity of the pumping profile as in present calculations or by other sources like natural disorder in the cavity. In a finite-size cavity, the domains with this wave vector form a ring-like structure along the border of the cavity, with a gas of mostly same-sign vortices in the center. The total angular momentum is imposed by the sign of TE-TM SOC, the wave vector at which the instability develops, and the cavity size. This effect can be detected experimentally via local dispersion measurements or by interference. The proposed configuration thus allows simultaneous experimental studies of quantum turbulence and high-angular-momentum states in continuously pumped exciton-polariton condensates.