This study aims to integrate existing measured data with a high-fidelity finite element (FE) model, thereby achieving a physics-based digital twin for PC girder bridges by employing a Bayesian model updating framework. The Fourier Neural Operator (FNO) is utilized as a surrogate model for Bayesian model selection and parameter estimation. Transitional Markov Chain Monte Carlo (TMCMC) sampling serves as the computational approach for estimating the posterior probability density functions (PDFs) of model parameters. The key idea behind FNO-aided Bayesian updates is to pre-train the FNO on a small dataset to approximate solutions of partial differential equations (PDEs) and then use many samples generated by the FNO to estimate the posterior PDFs of model parameters. The Gaussian random field and random variables as inputs ensure the subsequent employment of Bayesian model selection. The selection of a plausible model class is informed by the estimated model evidence. After implementing the updating framework, linear and nonlinear simulation results, including modal parameters and load-deflection curves obtained through the updated model, were found to be consistent with the experimentally obtained data. The most probable values (MPVs) derived from TMCMC sampling were then substituted into the FE model for simulation. Observations demonstrated that the load-deflection curve generated by solving the FE model matched perfectly with the curve estimated by the FNO. The FNO-aided Bayesian updating was more than 4000 times faster than conventional FE model updating. The accuracy and efficiency of the FNO-based surrogate model were verified simultaneously. Therefore, FNO-aided Bayesian updates can be applied in near real-time assessment of the structural performance of PC girder bridges in the field of model-based structural health monitoring (SHM).