The contribution of the box and crossed two-pion-exchange diagrams to proton-proton scattering at 90$^{\circ}_{c.m.}$ is calculated in the laboratory momentum range up to 12 GeV/c. Relativistic form factors related to the nucleon and pion size and representing the pion source distribution based on the quark structure of the nucleonic core are included at each vertex of the pion-nucleon interaction. Feynman diagram amplitudes calculated without form factors are checked against those derived from dispersion relations.