  We calculate the two-photon exchange corrections to electron-proton
scattering with nucleon and $\Delta$ intermediate states. The results show a
dependence on the elastic nucleon and nucleon-$\Delta$-transition form factors
used as input which leads to significant changes compared to previous
calculations. We discuss the relevance of these corrections and apply them to
the most recent and precise data set and world data from electron-proton
scattering. Using this, we show how the form factor extraction from these data
is influenced by the subsequent inclusion of physical constraints. The
determination of the proton charge radius from scattering data is shown to be
dominated by the enforcement of a realistic spectral function. Additionally,
the third Zemach moment from the resulting form factors is calculated. The
obtained radius and Zemach moment are shown to be consistent with Lamb shift
measurements in muonic hydrogen.
