The author investigate the Nevai operator \(N_n\) interpolating a function \(f\) on \([-1,1]\) in the more general weighted approximation case, when \(f\) may be unbounded at \(\pm 1\). It is shown that, similarly as for polynomials, the weighted convergence for the operators \(N_n\) with Pollaczek type weights is not guaranteed in general. Therefore the author considers weights vanishing algebraically at \(\pm 1\), i.e. functions having an algebraic singularity at \(\pm 1\). For such functions weighted uniform approximation estimates by \(N_n\) involving a suitable modulus of smoothness are given. Converse results are also established. Useful tools for these results are new weighted Markov-Bernstein inequalities for \(N_n\). It is shown that these results are sharp in some sense. When \(f\) is continuous on the whole interval \([-1,1]\), more precise direct and converse results, solving the saturation problem of \(N_n\) for \(s>2\), are obtained. Finally, the difficult problem of saturation of \(N_n\) for \(s\geqslant 2\) is investigated.