A search is presented for narrow resonances, with a mass between 0.6 and 1.8 TeV, decaying to pairs of jets, in proton-proton collisions at $\sqrt{s}=13$ TeV. The search is performed using dijets that are reconstructed, selected, and recorded in a compact form by the high-level trigger in a technique referred to as "data scouting", from data collected in 2016$-$18 corresponding to an integrated luminosity of 117 fb$^{-1}$. The dijet mass spectra are well described by a smooth parameterization, and no significant evidence for the production of new particles is observed. Model-independent upper limits are presented on the product of the cross section, branching fraction, and acceptance for the individual cases of narrow quark-quark, quark-gluon, and gluon-gluon resonances, and are compared to the predictions from a variety of models of narrow dijet resonance production. The upper limit on the coupling of a dark matter mediator to quarks is presented as a function of the mediator mass. The sensitivity of this search goes beyond what is expected from statistical scaling with luminosity alone, as a consequence of the use of fewer parameters in the background function within a more robust statistical procedure.