Although arrayed waveguide gratings (AWGs) are widely used in fibre Bragg grating (FBG) demodulation systems, their applications in real environments have been limited due to their narrow dynamic range and inability to continuously demodulate FBGs because of the finite bandwidth of AWG channels. Here, we developed a wide-dynamic-range, high-precision, continuous-demodulation FBG interrogator utilizing a dual-input channel on-chip silicon AWG. The introduction of two input channels in the AWG allowed two spectral peaks in each channel; therefore, staggered spectral peaks were realized. A joint-peak demodulation method based on this spectrum is proposed to improve the dynamic range and demodulation precision. With the proposed structure and method, we achieved continuous interrogation with a demodulation precision of 25.58 pm and a dynamic range of 24.5 nm in the 1537.5-1565.3 nm wavelength region. The relative demodulation accuracy within the full range reaches 0.1%. The dynamic range of adjacent channels is 5.4 nm, which is approximately 4 times greater than the dynamic range of an FBG conventional demodulation system using an AWG, and the relative demodulation accuracy is 0.47%. To our knowledge, the relative demodulation accuracy is currently the highest reported. This developed interrogator, with a core size of 420 µm × 300 µm, theoretically explained and experimentally verified the possibility of the accurate measurement of an arbitrary FBG wavelength with high demodulation accuracy in the measurement range. This work achieved continuous monitoring of external temperature by FBGs and demonstrated their significant potential in expanding the application field of FBGs.