We present an efficient tunable optical parametric generator (OPG) with its linewidth close to the Fourier transform limit by injection seeding a tunable diode laser. Benefitting from high-peak-power sub-nanosecond (426 ps) laser pumping and a high-gain MgO:PPLN (PPMgLN) crystal, the OPG produced signal peak power up to 0.343 MW at 1638 nm and the total conversion efficiency reached 47.9% at 1-kHz pulse repetition rate. Considering the linewidth limit of short signal pulses (∼ 350 ps), a tunable seeder with the linewidth at hundred-MHz level was applicable. The achieved OPG signal tuning range was 1510-1638 nm with linewidth at GHz level, which is two orders of magnitude narrower than the unseeded OPG. Injection seeding a non-resonant OPG device does not introduce extra cavity feedback electronics that are essential for an optical parametric oscillator (OPO), greatly improving robustness and reducing cost. It is believed such a compact, tunable and costless PPMgLN OPG with high peak power, high repetition rate and relatively narrow linewidth has great significance in lidar, spectroscopy, etc.