We obtain asymptotics of polynomials satisfying the orthogonality relations (Formula presented.) where the complex parameter (Formula presented.) is in the so-called two-cut region. As an application, we deduce asymptotic formulas for certain families of solutions of Painlevé-IV which are indexed by a nonnegative integer and can be written in terms of parabolic cylinder functions. The proofs are based on the characterization of orthogonal polynomials in terms of a Riemann–Hilbert problem and the Deift–Zhou nonlinear steepest descent method.