Owing to the inherent complementarity among LiDAR, camera, and IMU, a growing effort has been paid to laser-visual-inertial SLAM recently. The existing approaches, however, are limited in two aspects. First, at the front-end, they usually employ a discrete-time representation that requires high-precision hardware/software synchronization and are based on geometric laser features, leading to low robustness and scalability. Second, at the backend, visual loop constraints suffer from scale ambiguity and the sparseness of the point cloud deteriorates the scan-to-scan loop detection. To solve these problems, for the front-end, we propose a continuous-time laser-visual-inertial odometry which formulates the carrier trajectory in continuous time, organizes point clouds in probabilistic submaps, and jointly optimizes the loss terms of laser anchors, visual reprojections, and IMU readings, achieving accurate pose estimation even with fast motion or in unstructured scenes where it is difficult to extract meaningful geometric features. At the backend, we propose building 5-DoF laser constraints by matching projected 2D submaps and 6-DoF visual constraints via laser-aided visual relocalization, ensuring mapping consistency in large-scale scenes. Results show that our framework achieves high-precision estimation and is more robust than its counterparts when the carrier works in large scenes or with fast motion. The relevant codes and data are open-sourced at https://cslinzhang.github.io/Ct-LVI/Ct-LVI.html.