We revisit the approximation of nonempty compact planar sets by filled-in Julia sets of polynomials developed in [LinY] and analyze the rate of approximation. We use slightly modified fundamental Lagrange interpolation polynomials and show that taking certain classes of nodes with subexponential growth of Lebesgue constants 2010 Mathematics Subject Classification: Primary 30E10; Secondary 30C10, 30C85, 31A15, 37F10.