  The spatial distribution of charge and magnetization within the proton is
encoded in the elastic form factors. These have been precisely measured in
elastic electron scattering, and the combination of proton and neutron form
factors allows for the separation of the up- and down-quark contributions. In
this work, we extract the proton and neutron form factors from world's data
with an emphasis on precise new data covering the low-momentum region, which is
sensitive to the large-scale structure of the nucleon. From these, we separate
the up- and down-quark contributions to the proton form factors. We combine
cross section and polarization measurements of elastic electron-proton
scattering to separate the proton form factors and two-photon exchange (TPE)
contributions. We combine the proton form factors with parameterization of the
neutron form factor data and uncertainties to separate the up- and down-quark
contributions to the proton's charge and magnetic form factors. The extracted
TPE corrections are compared to previous phenomenological extractions, TPE
calculations, and direct measurements from the comparison of electron and
positron scattering. The flavor-separated form factors are extracted and
compared to models of the nucleon structure. With the inclusion of the precise
new data, the extracted TPE contributions show a clear change ofsign at low
$Q^2$, necessary to explain the high-$Q^2$ form factor discrepancy while being
consistent with the known $Q^2 \to 0$ limit. We find that the new Mainz data
yield a significantly different result for the proton magnetic form factor and
its flavor-separated contributions. We also observe that the RMS radius of both
the up- and down-quark distributions are smaller than the RMS charge radius of
the proton.
