Despite recent advances in circuit engineering, the design of genetic networks in mammalian cells is still painstakingly slow and fraught with inexplicable failures. Here, we demonstrate that transiently expressed genes in mammalian cells compete for limited transcriptional and translational resources. This competition results in the coupling of otherwise independent exogenous and endogenous genes, creating a divergence between intended and actual function. Guided by a resource-aware mathematical model, we identify and engineer natural and synthetic miRNA-based incoherent feedforward loop (iFFL) circuits that mitigate gene expression burden. The implementation of these circuits features the use of endogenous miRNAs as elementary components of the engineered iFFL device, a versatile hybrid design that allows burden mitigation to be achieved across different cell-lines with minimal resource requirements. This study establishes the foundations for context-aware prediction and improvement of in vivo synthetic circuit performance, paving the way towards more rational synthetic construct design in mammalian cells.