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We study the streaming instability of GeV$-100~$GeV cosmic rays (CRs) and its damping in the turbulent interstellar medium (ISM). We find that the damping of streaming instability is dominated by ion-neutral collisional damping in weakly ionized molecular clouds, turbulent damping in the highly ionized warm medium, and nonlinear Landau damping in the Galactic halo. Only in the Galactic halo, is the streaming speed of CRs close to the Alfv\'{e}n speed. Alfv\'{e}nic turbulence plays an important role in both suppressing the streaming instability and regulating the diffusion of streaming CRs via magnetic field line tangling, with the effective mean free path of streaming CRs in the observer frame determined by the Alfv\'{e}nic scale in super-Alfv\'{e}nic turbulence. The resulting diffusion coefficient is sensitive to Alfv\'{e}n Mach number, which has a large range of values in the multi-phase ISM. Super-Alfv\'{e}nic turbulence contributes to additional confinement of streaming CRs, irrespective of the dominant damping mechanism.