We establish sufficient conditions for the convergence of the Fourier expansions of functions from $L_\mu^p$ ($p\geqslant1$) in terms of the order of growth of the system $\{\varphi_n(t)\}$, of polynomial kind, orthonormal with respect to the measure $\mu(t)$ on $[a, b]$ and containing a constant. The convergence is considered either in a given point of the orthogonality interval or inside the interval $[c,d]\subset[a,b]$. In connection with this we obtain estimates for the Lebesgue functions of the system $\{\varphi_n(t)\}$, and we consider the localization problem of the convergence conditions.