Nonenzymatic catalytic substrates have been engineered using toehold-mediated DNA strand displacement, and their programmable applications range from medical diagnosis to molecular computation. However, the complexity, stability, scalability, and sensitivity of those systems are plagued by network leakage. A novel way to suppress leakage is to increase its energy barrier through four-way branch migration. Presented here, we designed multi-arm junction substrates that simultaneously exploit four-way branch migration, with a high-energy barrier to minimize leakage, and three-way branch migration, with a low-energy barrier to maximize catalysis. Original feed forward, autocatalytic, and cross-catalytic systems have been designed with polynomial and exponential amplification that exhibit the modularity of linear substrates and the stability of hairpin substrates, creating a new phase space for synthetic biologist, biotechnologist, and DNA nanotechnologists to explore. A key insight is that high-performing circuits can be engineered in the absence of intensive purification and/or extensive rounds of design optimization. Without adopting established leakage suppression techniques, the ratio of the catalytic rate constant to the leakage rate constant is more than 2 orders of magnitude greater than state-of-the-art linear and hairpin substrates. Our results demonstrate that multi-arm junctions have great potential to become central building blocks in dynamic DNA nanotechnology.