Understanding the normal state is essential for uncovering the mechanism of high‑$T_c$ superconductivity. We investigate magnetotransport in Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ and Bi${}_2$Sr${}_2$Ca${}_2$Cu${}_3$O${}_{10+\delta }$ single crystals over a wide doping range. While the in‑plane resistivity and Hall coefficient show strong pseudogap‑induced temperature dependence, the $T^2$ Hall-angle behavior and the modified Kohler’s rule remain robust across all dopings. The onset temperatures of the pseudogap are clearly distinct from superconducting fluctuations, although they scale with the pseudogap magnitudes with a factor consistent with a $d$-wave superconductor. These results demonstrate that the pseudogap does not arise from superconducting fluctuations and instead suggest that it may originate from preformed Cooper pairing in the BCS–BEC crossover regime.