Simultaneous localization and mapping (SLAM) is considered a challenge in environments with many moving objects. This paper proposes a novel LiDAR inertial odometry framework, LiDAR inertial odometry-based on indexed point and delayed removal strategy (ID-LIO) for dynamic scenes, which builds on LiDAR inertial odometry via smoothing and mapping (LIO-SAM). To detect the point clouds on the moving objects, a dynamic point detection method is integrated, which is based on pseudo occupancy along a spatial dimension. Then, we present a dynamic point propagation and removal algorithm based on indexed points to remove more dynamic points on the local map along the temporal dimension and update the status of the point features in keyframes. In the LiDAR odometry module, a delay removal strategy is proposed for historical keyframes, and the sliding window-based optimization includes the LiDAR measurement with dynamic weights to reduce error from dynamic points in keyframes. We perform the experiments both on the public low-dynamic and high-dynamic datasets. The results show that the proposed method greatly increases localization accuracy in high-dynamic environments. Additionally, the absolute trajectory error (ATE) and average RMSE root mean square error (RMSE) of our ID-LIO can be improved by 67% and 85% in the UrbanLoco-CAMarketStreet dataset and UrbanNav-HK-Medium-Urban-1 dataset, respectively, when compared with LIO-SAM.