We have carried out Raman spectroscopy experiments to investigate two-magnon excitations in epitaxial thin films of the quasi-two-dimensional antiferromagnetic Mott insulator Sr$_2$IrO$_4$ under in-plane misfit strain. With in-plane biaxial compression, the energy of the two-magnon peak increases, and the peak remains observable over a wider temperature range above the Neel temperature, indicating strain-induced enhancement of the superexchange interactions between $it{J}_{eff}$ = 1/2 pseudospins. From density functional theory calculations, we have found an increase of the nearest-neighbor hopping parameter and exchange interaction with increasing biaxial compressive strain, in agreement with the experimental observations. Our experimental and theoretical results provide perspectives for systematic, theory-guided strain control of the primary exchange interactions in 5$it{d}$ transition metal oxides.
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