Defect-tolerant perovskite nanocrystals of the general formula Cs-Pb-$X_3$ (where $X=\mathrm{Cl}$, Br, and I) have shown exceptional potential in fundamental physics as well as in novel optoelectronic applications as the next generation of solar cells. Although exciton many-body interactions such as biexciton Stark shift, state filling, and Auger recombination are studied extensively, other important correlated effects, such as band gap renormalization (BGR) and hot-phonon bottleneck, are not explored in these nanocrystals. Here we experimentally demonstrate the carrier density dependence of the BGR and an effective hot-phonon bottleneck in $\mathrm{CsPb}{\left(\mathrm{C}{\mathrm{l}}_{0.20}\mathrm{B}{\mathrm{r}}_{0.80}\right)}_3$ mixed-halide nanocrystals. The results are compared with two other halide compositions, namely, ${\mathrm{CsPbBr}}_3$ and $\mathrm{CsPb}{\left(\mathrm{B}{\mathrm{r}}_{0.55}{\mathrm{I}}_{0.45}\right)}_3$ nanocrystals with varying band gaps. The optical response of the nanocrystals changes dramatically across the spectral range of many hundreds of meV at high carrier density due to large BGR. We have calculated the BGR constant $\approx \left(6.0\pm 0.3\right)\times {10}^{-8}\mathrm{eV}$ cm for $\mathrm{CsPb}{\left(\mathrm{C}{\mathrm{l}}_{0.20}\mathrm{B}{\mathrm{r}}_{0.80}\right)}_3$ nanocrystals that provides the amount of band gap shift as a function of carrier density. In these nanocrystals, an efficient hot-phonon bottleneck is observed at a carrier density of $3.1\times {10}^{17}\mathrm{c}{\mathrm{m}}^{-3}$ that slows down the thermalization by 1 order of magnitude. Our findings reveal that the complex kinetic profile of the exciton dynamics can be analyzed by the global target analysis using the sequential model with increasing lifetimes.