This paper is concerned with bifurcation of limit cycles in a fourth-order near-Hamiltonian system with quartic perturbations. By bifurcation theory, proper perturbations are given to show that the system may have 20, 21 or 23 limit cycles with different distributions. This shows thatH(4) ≥ 20, whereH(n) is the Hilbert number for the second part of Hilbert’s 16th problem. It is well known that H(2) ≥ 4, and it has been recently proved that H(3) ≥ 12. The number of limit cycles obtained in this paper greatly improves the best existing result, H(4) ≥ 15, for fourth-degree polynomial planar systems.