Searches are conducted for new spin-0 or spin-1 bosons using events where a Higgs boson with mass 125 GeV decays into four leptons (ℓ = e, μ). This decay is presumed to occur via an intermediate state which contains two on-shell, promptly decaying bosons: H → XX/ZX → 4ℓ, where the new boson X has a mass between 1 and 60 GeV. The search uses pp collision data collected with the ATLAS detector at the LHC with an integrated luminosity of 139 fb⁻¹ at a centre-of-mass energy s = 13 TeV. The data are found to be consistent with Standard Model expectations. Limits are set on fiducial cross sections and on the branching ratio of the Higgs boson to decay into XX/ZX, improving those from previous publications by a factor between two and four. Limits are also set on mixing parameters relevant in extensions of the Standard Model containing a dark sector where X is interpreted to be a dark boson. [Figure not available: see fulltext.]