Charged-current neutrino-nucleus scattering is studied in the quasielastic region with various relativistic nuclear models that include the relativistic Hartree, the nonlinear $\sigma$, the quark-meson coupling, and the chiral quark-meson coupling models. Theoretical results at several kinematics are compared with experimental data measured from MiniBooNE, T2K, MicroBooNE, and SciBooNE. While the theoretical double-differential cross sections in terms of the kinetic energy and polar angle of outgoing muon describe the T2K data well, the single-differential cross sections in terms of energy loss and the total cross sections of incident neutrino energy do not properly describe MiniBooNE, MicroBooNE, and SciBooNE data. Additionally, differences from the relativistic nuclear models are examined in detail by comparing with the experimental data. In particular, the MiniBooNE data are not consistent with MicroBooNE and SciBooNE data, where the incident neutrino energies are below 1 GeV.