The cost and effort of modelling existing bridges from point clouds currently outweighs the perceived benefits of the resulting model. There is a pressing need to automate this process. Previous research has achieved the automatic generation of surface primitives combined with rule-based classification to create labelled cuboids and cylinders from point clouds. While these methods work well in synthetic datasets or idealized cases, they encounter huge challenges when dealing with real-world bridge point clouds, which are often unevenly distributed and suffer from occlusions. In addition, real bridge geometries are complicated. In this paper, we propose a novel top-down method to tackle these challenges for detecting slab, pier, pier cap, and girder components in reinforced concrete bridges. This method uses a slicing algorithm to separate the deck assembly from pier assemblies. It then detects and segment pier caps using their surface normal, and girders using oriented bounding boxes and density histograms. Finally, our method merges over-segments into individually labelled point clusters. The results of 10 real-world bridge point cloud experiments indicate that our method achieves an average detection precision of 98.8%. This is the first method of its kind to achieve robust detection performance for the four component types in reinforced concrete bridges and to directly produce labelled point clusters. Our work provides a solid foundation for future work in generating rich Industry Foundation Classes models from the labelled point clusters.