Large long-distance standard model effects in flavor-changing neutral current (FCNC) semileptonic $D$ decays can make observable these processes in future measurements. Eventual disagreements in this sector and/or the observation of lepton-family-violating (LFV) $D$ decays would require an explanation beyond the standard model framework. In this paper we confront present experimental data on leptonic and semileptonic FCNC and LFV $D$ meson decays with a version of the two-Higgs-doublet model that allows these effects to occur at the tree level. The stringent bounds on the parameters of the model are obtained from ${D}^{0}\ensuremath{\rightarrow}{l}^{+}{l}^{{\ensuremath{'}}^{}\ensuremath{-}}$ and $\stackrel{\ensuremath{\rightarrow}}{D}\ensuremath{\pi}{l}^{+}{l}^{{\ensuremath{'}}^{}\ensuremath{-}}$ decays. The consistency of the model requires that the branching fractions of $\stackrel{\ensuremath{\rightarrow}}{D}{\mathrm{Vl}}^{+}{l}^{{\ensuremath{'}}^{}\ensuremath{-}}$ decays should be below the ${10}^{\ensuremath{-}9}$ level.