We study $WW$ and $WZ$ production with $\ensuremath{\ell}\ensuremath{\nu}qq$ ($\ensuremath{\ell}=e,\ensuremath{\mu}$) final states using data collected by the D0 detector at the Fermilab Tevatron Collider corresponding to $4.3\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of integrated luminosity from $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$. Assuming the ratio between the production cross sections $\ensuremath{\sigma}(WW)$ and $\ensuremath{\sigma}(WZ)$ as predicted by the standard model, we measure the total $WV$ ($V=W,Z$) cross section to be $\ensuremath{\sigma}(WV)={19.6}_{\ensuremath{-}3.0}^{+3.2}\text{ }\text{ }\mathrm{pb}$ and reject the background-only hypothesis at a level of 7.9 standard deviations. We also use $b$-jet discrimination to separate the $WZ$ component from the dominant $WW$ component. Simultaneously fitting $WW$ and $WZ$ contributions, we measure $\ensuremath{\sigma}(WW)={15.9}_{\ensuremath{-}3.2}^{+3.7}\text{ }\text{ }\mathrm{pb}$ and $\ensuremath{\sigma}(WZ)={3.3}_{\ensuremath{-}3.3}^{+4.1}\text{ }\text{ }\mathrm{pb}$, which is consistent with the standard model predictions.