A search for light long-lived particles decaying to displaced jets is presented, using a data sample of proton-proton collisions at a center-of-mass energy of $13.6~\mathrm{TeV}$, corresponding to an integrated luminosity of $34.7~\mathrm{fb^{-1}}$, collected with the CMS detector at the CERN LHC in 2022. Novel trigger, reconstruction, and machine-learning techniques were developed for and employed in this search. After all selections, the observations are consistent with the background predictions. Limits are presented on the branching fraction of the Higgs boson to long-lived particles that subsequently decay to quark pairs or tau lepton pairs. Up to a factor of 10 improvement is achieved over previous limits for models with long-lived particle masses smaller than $60~\mathrm{GeV}$ and proper decay lengths smaller than $1~\mathrm{m}$. The first constraints are placed on the fraternal twin Higgs and folded supersymmetry models, where the lower bounds on the top quark partner mass reach up to $350~\mathrm{GeV}$ for the fraternal twin Higgs model and $250~\mathrm{GeV}$ for the folded supersymmetry model.