We present a self-interaction-free time-dependent density-functional theory (TDDFT) for nonperturbative treatment of multiphoton processes of many-electron molecular systems in intense laser fields. The time-dependent exchange-correlation (xc) energy potential with proper short- and long-range potential is constructed by means of the time-dependent optimized effective potential (OEP) method and the incorporation of an explicit self-interaction-correction (SIC) term. The resulting time-dependent OEP/SIC equations are structurally similar to the time-dependent Hartree-Fock equations, but include the many-body effects through an orbital-independent single-particle local time-dependent xc potential. A numerical time-propagation technique is introduced for accurate and efficient solution of the TDDFT/OEP-SIC equations for two-center diatomic molecular systems. This procedure involves the use of a generalized pseudospectral method for nonuniform optimal grid discretization of the Hamiltonian in prolate spheroidal coordinates and a split-operator scheme in the energy representation for the time development of the electron orbital wave functions. High-precision time-dependent wave functions can be obtained by this procedure with the use of only a modest number of spatial grid points. The theory is applied to a detailed study of high-order harmonic generation (HHG) processes of ${\mathrm{H}}_{2}$ molecules in intense pulsed laser fields. Particular attention is paid to the exploration of the spectral and temporal structures of HHG by means of the wavelet time-frequency analysis. The results reveal striking details of the spectral and temporal fine structures of HHG, providing new insights regarding the detailed HHG mechanisms in different energy regimes.