In this paper, new strategies are devised for joint load balancing and interference management in the downlink of a heterogeneous network, where small cells are densely deployed within the coverage area of a traditional macrocell. Unlike existing work, the limited backhaul capacity at each base station (BS) is taken into account. Here users (UEs) cannot be offloaded to any arbitrary BS, but only to ones with sufficient backhaul capacity remaining. Jointly designed with traffic offload, transmit power allocation mitigates the intercell interference to further support the quality of service of each UE. The objective here is either (i) to maximize the network sum rate subject to minimum throughput requirements at individual UEs, or (ii) to maximize the minimum UE throughput. Both formulated problems belong to the difficult class of mixed-integer nonconvex optimization. The inherently binary BS-UE association variables are strongly coupled with the transmit power variables, making the problems even more challenging to solve. New iterative algorithms are developed based on an exact penalty method combined with successive convex programming, where we deal with the binary BS-UE association problem and the nonconvex power allocation problem one at a time. At each iteration of the proposed algorithms, only two simple convex problems need to be solved in the same time scale. It is proven that the algorithms improve the objective functions in each iteration and converge eventually. Numerical results demonstrate the efficiency of the proposed algorithms in both traffic offloading and interference mitigation.