We theoretically study the phase relaxation of excitons in doped single-walled carbon nanotubes caused by elastic scattering between an exciton and an electron. Our calculation of the spectral linewidth, that is the inverse of the scattering rate, demonstrates that this dephasing process is unexpectedly suppressed in doped single-walled carbon nanotubes, giving rise to anomalous nonlinear behavior as a function of the electron density. A detailed investigation reveals that the logarithmic correction of the exciton self-energy plays a critical role in the exciton-carrier scattering dynamics.