A search for new-physics resonances decaying into a lepton and a jet performed by the ATLAS experiment is presented. Scalar leptoquarks pair-produced in $pp$ collisions at $\sqrt{s}=13$ TeV at the Large Hadron Collider are considered using an integrated luminosity of 139 fb$^{-1}$, corresponding to the full Run 2 dataset. They are searched for in events with two electrons or two muons and two or more jets, including jets identified as arising from the fragmentation of $c$- or $b$-quarks. The observed yield in each channel is consistent with the Standard Model background expectation. Leptoquarks with masses below 1.8 TeV and 1.7 TeV are excluded in the electron and muon channels, respectively, assuming a branching ratio into a charged lepton and a quark of 100%, with minimal dependence on the quark flavour. Upper limits on the aforementioned branching ratio are also given as a function of the leptoquark mass.