Using a sample of about 10^{10} J/ψ events collected at a center-of-mass energy sqrt[s]=3.097 GeV with the BESIII detector, the electromagnetic Dalitz decays J/ψ→e^{+}e^{-}π^{+}π^{-}η^{'}, with η^{'}→γπ^{+}π^{-} and η^{'}→π^{+}π^{-}η, have been studied. The decay J/ψ→e^{+}e^{-}X(1835) is observed with a significance of 15σ, and also an e^{+}e^{-} invariant-mass dependent transition form factor of J/ψ→e^{+}e^{-}X(1835) is presented for the first time. The intermediate states X(2120) and X(2370) are also observed in the π^{+}π^{-}η^{'} invariant-mass spectrum with significances of 5.3σ and 7.3σ. The corresponding product branching fractions for J/ψ→e^{+}e^{-}X, X→π^{+}π^{-}η^{'} [X=X(1835), X(2120), and X(2370)] are reported.