We present new precision measurements of the elastic electron-proton scattering cross section for momentum transfer (Q2) up to 15.75(GeV/c)2. Combined with existing data, these provide an improved extraction of the proton magnetic form factor at high Q2 and double the range over which a longitudinal or transverse separation of the cross section can be performed. The difference between our results and polarization data agrees with that observed at lower Q2 and attributed to hard two-photon exchange (TPE) effects, extending to 8(GeV/c)2 the range of Q2 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 Q2.