In this work, we discuss exclusive semileptonic $B_c$-meson decays: $B_cto eta_c(J/psi)l u$ and $B_cto D(D^*)l u$ in the framework of the relativistic independent quark(RIQ) model based on an average flavor independent confining potential in equally mixed scalar-vector harmonic form. We calculate the invariant form factors representing decay amplitudes from the overlapping integrals of meson wave functions derivable in the RIQ model. To evaluate the lepton mass effects in the semileptonic decays, we first study the $q^2$-dependence of the form factors in the accessible kinematic range of $q^2$ involved in the decay process in its $e^-$ and $tau^-$ mode separately. Similar studies on helicity amplitudes, $q^2-$spectra for different helicity contributions, and total $q^2$-spectra for each decay process are carried out separately in their $e^-$ and $tau^-$ modes. We predict the decay rates/ branching fractions, forward-backward asymmetry, and the asymmetry parameter in reasonable agreement with other model predictions, which can hopefully be tested in future experiments at the Tevatron and LHC. We also predict the observable $R$ which corresponds to the ratio of branching fractions for the decay process in its $e^-$ mode to its corresponding value in the $tau^-$ mode. Our results are comparable to another standard model(SM) predictions which highlight the failure of the lepton flavor universality hinting at new physics beyond SM for the explanation of the observed deviation of observable $R$ value from the corresponding SM predictions.
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