  We present new precision measurements of the elastic electron-proton
scattering cross section for momentum transfer (Q$^2$) up to 15.75~\gevsq.
Combined with existing data, these provide an improved extraction of the proton
magnetic form factor at high Q$^2$ and double the range over which a
longitudinal/transverse separation of the cross section can be performed. The
difference between our results and polarization data agrees with that observed
at lower Q$^2$ and attributed to hard two-photon exchange (TPE) effects,
extending to 8~(GeV/c)$^2$ the range of Q$^2$ for which a discrepancy is
established at $>$95\% confidence. We use the discrepancy to quantify the size
of TPE contributions needed to explain the cross section at high Q$^2$.
