Jets with different radius parameters R are an important tool for probing quantum chromodynamics processes at different angular scales. Jets with small $R=0.2$ are instrumental in measurements of the substructure of large-R jets resulting from collimated hadronic decays of energetic W, Z, and Higgs bosons, top quarks, and of potential new resonances. This paper presents measurements of the energy scale, resolution, and associated uncertainties of jets with radius parameters $R=0.2$ and 0.6, obtained using the ATLAS detector. The results are based on $37~\text {fb}^{-1}$ of proton–proton collision data from the Large Hadron Collider at a centre-of-mass energy of $\sqrt{s}=13$ TeV. A new in situ method for measuring jet energy scale differences between data and Monte Carlo simulations is presented. The systematic uncertainties in the jet energy scale for central jets $(|\eta | < 1.2)$ typically vary from 1% to about 5% as a function of $|\eta |$ at very low transverse momentum, $p_{\text {T}},$ of around 20 GeV for both $R=0.2$ and 0.6 jets. The relative energy resolution ranges from $(35 \pm 6)\%$ at $p_{\text {T}} = 20$ GeV to $(6 \pm 0.5)\%$ at $p_{\text {T}} = 300$ GeV for central $R=0.2$ jets, and is found to be slightly worse for $R=0.6$ jets. Finally, the effect of close-by hadronic activity on the jet energy scale is investigated and is found to be well modelled by the ATLAS Monte Carlo simulations.