We present results on the nucleon electromagnetic form factors from Lattice QCD at momentum transfer up to about $12$ GeV$^2$. We analyze two gauge ensembles with the Wilson-clover fermion action, a lattice spacing of $a\approx 0.09$ fm and pion masses $m_\pi\approx 170$ MeV and $m_\pi\approx 280$ MeV. In our analysis we employ momentum smearing as well as a set of techniques to investigate excited state effects. Good agreement with experiment and phenomenology is found for the ratios $G_E/G_M$ and $F_2/F_1$, whereas discrepancies are observed for the individual form factors $F_1$ and $F_2$. We discuss various systematics that may affect our calculation.