This paper presents a measurement of quantities related to the formation of jets from high-energy quarks and gluons (fragmentation). Jets with transverse momentum 100 GeV 2.5 TeV and pseudorapidity from an integrated luminosity of 33 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of $s=13$ TeV proton-proton collisions are reconstructed with the ATLAS detector at the Large Hadron Collider. Charged-particle tracks with $pT>500$ MeV and are used to probe the detailed structure of the jet. The fragmentation properties of the more forward and the more central of the two leading jets from each event are studied. The data are unfolded to correct for detector resolution and acceptance effects. Comparisons with parton shower Monte Carlo generators indicate that existing models provide a reasonable description of the data across a wide range of phase space, but there are also significant differences. Furthermore, the data are interpreted in the context of quark- and gluon-initiated jets by exploiting the rapidity dependence of the jet flavor fraction. A first measurement of the charged-particle multiplicity using model-independent jet labels (topic modeling) provides a promising alternative to traditional quark and gluon extractions using input from simulation. The simulations provide a reasonable description of the quark-like data across the jet $pT$ range presented in this measurement, but the gluon-like data have systematically fewer charged particles than the simulations.