The singular value decomposition (SVD) is a crucial tool in machine learning and statistical data analysis. However, it is highly susceptible to outliers in the data matrix. Existing robust SVD algorithms often sacrifice speed for robustness or fail in the presence of only a few outliers. This study introduces an efficient algorithm, called Spherically Normalized SVD, for robust SVD approximation that is highly insensitive to outliers, computationally scalable, and provides accurate approximations of singular vectors. The proposed algorithm achieves remarkable speed by utilizing only two applications of a standard reduced-rank SVD algorithm to appropriately scaled data, significantly outperforming competing algorithms in computation times. To assess the robustness of the approximated singular vectors and their subspaces against data contamination, we introduce new notions of breakdown points for matrix-valued input, including row-wise, column-wise, and block-wise breakdown points. Theoretical and empirical analyses demonstrate that our algorithm exhibits higher breakdown points compared to standard SVD and its modifications. We empirically validate the effectiveness of our approach in applications such as robust low-rank approximation and robust principal component analysis of high-dimensional microarray datasets. Overall, our study presents a highly efficient and robust solution for SVD approximation that overcomes the limitations of existing algorithms in the presence of outliers.