A measurement is presented of electroweak (EW) production of a W boson in association with two jets in proton-proton collisions at $\sqrt{s}=$ 13 TeV. The data sample was recorded by the CMS Collaboration at the LHC and corresponds to an integrated luminosity of 35.9 fb$^{-1}$. The measurement is performed for the $\ell\nu$jj final state (with $\ell\nu$ indicating a lepton-neutrino pair, and j representing the quarks produced in the hard interaction) in a kinematic region defined by invariant mass $m_\mathrm{jj} > $ 120 GeV and transverse momenta $p_\mathrm{T j} >$ 25 GeV. The cross section of the process is measured in the electron and muon channels yielding $\sigma_\mathrm{EW}($Wjj$)=$ 6.23$\pm$0.12 (stat)$\pm$ 0.61 (syst) pb per channel, in agreement with leading-order standard model predictions. The additional hadronic activity of events in a signal-enriched region is studied, and the measurements are compared with predictions. The final state is also used to perform a search for anomalous trilinear gauge couplings. Limits on anomalous trilinear gauge couplings associated with dimension-six operators are given in the framework of an effective field theory. The corresponding 95% confidence level intervals are $-$2.3 $< c_{\mathrm{WWW}}/\Lambda^2 <$ 2.5 TeV$^{-2}$, $-$8.8 $< c_{\mathrm{W}}/\Lambda^2 <$ 16 TeV$^{-2}$, and $-$45 $< c_{\mathrm{B}}/\Lambda^2 <$ 46 TeV$^{-2}$. These results are combined with the CMS EW Zjj analysis, yielding the most stringent limit to date on the $c_{\mathrm{WWW}}$ coupling: $-$1.8 $< c_{\mathrm{WWW}}/\Lambda^2 <$ 2.0 TeV$^{-2}$.