Context: Alfvén waves (AWs) excited by the cosmic-ray (CR) streaming instability (CRSI) are a fundamental ingredient for CR confinement. The effectiveness of such self-confinement relies on a balance between the CRSI growth rate and the damping mechanisms acting on quasi-parallel AWs excited by CRs. One relevant mechanism is called turbulent damping, in which an AW packet injected in pre-existing turbulence undergoes a cascade process due to its nonlinear interaction with fluctuations of the background. Aims: The turbulent damping of an AW packet in pre-existing magnetohydrodynamic (MHD) turbulence is re-examined, revised, and extended to include the most recent theories of MHD turbulence that account for dynamic alignment and reconnection-mediated regimes. The case in which the role of feedback of CR-driven AWs on pre-existing turbulence is important is also discussed. Methods. The Elsässer formalism is employed. Particular attention is given to the role of a nonlinearity parameter χw that estimates the strength of the nonlinear interaction between CR-driven AW packets and the background fluctuations. We point out the difference between χw and the parameter χz that instead describes the intrinsic strength of nonlinear interactions between pre-existing fluctuations. Turbulent damping rates of quasi-parallel AW packets and cosmic-ray feedback (CRF) are derived within this formalism. Results: When the strength of the nonlinear interaction is properly taken into account, we find that (i) the turbulent damping rate of quasi-parallel AWs in sub-Alfvénic turbulence depends on the background-fluctuation amplitude to the third power, and hence is strongly suppressed; (ii) the dependence on the AW’s wavelength (and thus on the CR gyro-radius from which it is excited) is different from what has been previously obtained; and (iii) when dynamic alignment of cascading fluctuations and the possibility of a reconnection-mediated range is included in the picture, the turbulent damping rate exhibits novel regimes and breaks. Finally, a criterion for CRF is derived and a simple phenomenological model of CR-modified scaling of background fluctuations is provided.