Measurements of the substructure of top-quark jets are presented, using Formula Presented of 13 TeV Formula Presented collision data recorded with the ATLAS detector at the LHC. Top-quark jets reconstructed with the anti-Formula Presented algorithm with a radius parameter Formula Presented are selected in top-quark pair (Formula Presented) events where one top quark decays semileptonically and the other hadronically, or where both top quarks decay hadronically. The top-quark jets are required to have transverse momentum Formula Presented, yielding large samples of data events with jet Formula Presented values between 350 and 600 GeV. One- and two-dimensional differential cross sections for eight substructure variables, defined using only the charged components of the jets, are measured in a particle-level phase space by correcting for the smearing and acceptance effects induced by the detector. The differential cross sections are compared with the predictions of several Monte Carlo simulations in which top-quark pair-production quantum chromodynamic matrix-element calculations at next-to-leading-order precision in the strong coupling constant Formula Presented are passed to leading-order parton shower and hadronization generators. The Monte Carlo predictions for measures of the broadness, and also the two-body structure, of the top-quark jets are found to be in good agreement with the measurements, while variables sensitive to the three-body structure of the top-quark jets exhibit some tension with the measured distributions.