With the scale of antenna arrays and the bandwidth increasing, many existing narrowband channel estimation methods ignoring the effect of beam squint may face severe performance degradation in wideband millimeter-wave (mmWave) communication systems. In this letter, a wideband Newtonized orthogonal matching pursuit (wNOMP) algorithm has been proposed to perform channel estimation. The proposed method based on the minimum mean square error (MMSE) criterion is optimal for Gaussian noise. Considering real communication systems, it is common that the noise follows a non-Gaussian distribution. Accordingly we extend the wideband channel estimation method via the minimum $\ell_p$-norm criterion which enhances the robustness against the non-Gaussian noise. Simulations have been conducted to validate the superiority of the proposed method over other representative methods.