Measurements of Higgs boson production in the off-shell region are presented, using the four-lepton decay channel. Data from proton-proton collisions at the CERN LHC, collected by the CMS experiment and corresponding to an integrated luminosity of 138 fb${}^{-1}$ at a center-of-mass energy of 13 TeV, are utilized. The first direct test of composite Higgs boson models is performed, with a lower limit on the compositeness scale ${\Lambda }_H$ set at 870 GeV at the 95% confidence level. Tests of gluon-fusion production within the standard model effective field theory framework are performed. The first constraint on the Higgs boson self-coupling in the off-shell region is obtained. By combining on- and off-shell measurements, the analysis sets the tightest constraints to date on light-quark Yukawa couplings, while relaxing assumptions such as the bound on the Higgs boson coupling to vector bosons, thereby providing more model-independent results. Constraints on the Higgs boson width are provided while accounting for a range of beyond-the-standard-model effects, including both light and heavy particles in the gluon-fusion production loop, as well as modified couplings to vector bosons. A combined analysis of the H$\to$ZZ and PH$\to$WW channels is performed to improve sensitivity to the Higgs boson width, yielding ${\Gamma }_H$ = 5.1${}_{-1.8}^{+2.0}$ MeV. The scenario of no off-shell Higgs boson production is excluded at a confidence level exceeding 5 standard deviations.