Condensation of water vapor on active cloud condensation nuclei produces micron-size water droplets. To form rain, they must grow rapidly into at least 50- to 100-μm droplets. Observations show that this process takes only 15-20 min. The unexplained physical mechanism of such fast growth is crucial for understanding and modeling of rain and known as "condensation-coalescence bottleneck in rain formation." We show that the recently discovered phenomenon of the tangling clustering instability of small droplets in temperature-stratified turbulence [Phys. Fluids 25, 085104 (2013)] results in the formation of droplet clusters with drastically increased droplet number densities. The mechanism of the tangling clustering instability is much more effective than the previously considered by us the inertial clustering instability caused by the centrifugal effect of turbulent vortices. This is the reason of strong enhancement of the collision-coalescence rate inside the clusters. The mean-field theory of the droplet growth developed in this study can be useful for explanation of the observed fast growth of cloud droplets in warm clouds from the initial 1-μm-size droplets to 40- to 50-μm-size droplets within 15-20 min.