  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 N\'eel
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.
