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 ($\ell =$ $e$,$\mu$). This decay is presumed to occur via an intermediate state which contains two on-shell, promptly decaying bosons: $H\to XX/ZX\to 4\ell$, 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$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ 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.