It is challenging to develop numerical methods that simultaneously maintain accuracy of resolving boundary conditions and mesh flexibility to handle the interface in interface-coupled multi-physics problems involving large property discontinuity. On the one hand, boundary-fitted methods possess high accuracy in capturing the interfacial phenomenon but involve complicated volumetric mesh generation, mesh-motion, and even re-meshing procedures. On the other hand, immersed boundary methods (IBM) provide mesh flexibility but sometimes suffer from inferior interface representations, leading to poor enforcement of boundary conditions. This talk will present an enriched immersed boundary method (EIBM) to overcome this challenge and demonstrates its efficacy in conjugate heat transfer, a representative example in interface-coupled multi-physics systems that have implications for many industrial processes. The core technique of the method is to enhance IBM's accuracy of the fluid-solid interface by enriching the degrees of freedom (DoFs) of the cut elements to enforce temperature and flux compatibilities and resolve all the physical unknowns on the background mesh to simplify the volumetric mesh generation. We implement the EIBM under the framework of a variational multiscale formulation for coupled Navier-Stokes and thermodynamics equations. The enriched DoFs enable better enforcement of temperature and flux compatibilities with large conductivity ratios across the fluid-solid interface. At the same time, the immersed nature of the proposed method still attains mesh flexibility. The EIBM's accuracy is thoroughly evaluated through a set of examples, ranging from benchmark problems with analytical solutions to real-world cooling processes of a moving metallic structure with complex geometry.