Thin-film lithium niobate is a promising material platform for integrated nonlinear photonics, due to its high refractive index contrast with the excellent optical properties. However, the high refractive index contrast and correspondingly small mode field diameter limit the attainable coupling between the waveguide and fiber. In second harmonic generation processes, lack of efficient fiber-chip coupling schemes covering both the fundamental and second harmonic wavelengths has greatly limited the overall efficiency. We design and fabricate an ultra-broadband tri-layer edge coupler with a high coupling efficiency. The coupler allows efficient coupling of 1  dB  /  facet at 1550 nm and 3  dB  /  facet at 775 nm. This enables us to achieve an ultrahigh overall second harmonic generation normalized efficiency (fiber-to-fiber) of 1027  %    W^−  1 cm^−  2 (on-chip second harmonic efficiency ∼3256  %    W^−  1 cm^−  2) in a 5-mm-long periodically-poled lithium niobate waveguide, which is two to three orders of magnitude higher than that in state-of-the-art devices.