Recovering a low rank matrix from a subset of its entries, some of which may be corrupted, is known as the robust matrix completion (RMC) problem. Existing RMC methods have several limitations: they require a relatively large number of observed entries; they may fail under overparametrization, when their assumed rank is higher than the correct one; and many of them fail to recover even mildly ill-conditioned matrices. In this paper we propose a novel RMC method, denoted $\texttt{RGNMR}$, which overcomes these limitations. $\texttt{RGNMR}$ is a simple factorization-based iterative algorithm, which combines a Gauss-Newton linearization with removal of entries suspected to be outliers. On the theoretical front, we prove that under suitable assumptions, $\texttt{RGNMR}$ is guaranteed exact recovery of the underlying low rank matrix. Our theoretical results improve upon the best currently known for factorization-based methods. On the empirical front, we show via several simulations the advantages of $\texttt{RGNMR}$ over existing RMC methods, and in particular its ability to handle a small number of observed entries, overparameterization of the rank and ill-conditioned matrices.