We study an extension of the Twin Higgs model that solves the Hierarchy problem and simultaneously addresses problems of the large- and small-scale structures of the Universe. Besides naturally providing dark matter (DM) candidates as the lightest charged twin fermions, the twin sector contains a light photon and neutrinos, which can modify structure formation relative to the prediction from the $Λ$CDM paradigm. We focus on a Fraternal Twin Higgs scenario in which the spin-3/2 baryon $\hat{\Omega}∼(\hat{b}\hat{b}\hat{b})$ and the lepton twin tau $\hat{\tau}$ contribute to the dominant and subcomponent dark matter densities. A non-decoupled scattering between the twin tau and twin neutrino arising from a gauged twin lepton number symmetry provides a drag force that damps the density inhomogeneity of a dark matter subcomponent. This realizes the Partially Acoustic dark matter (PAcDM) scenario and explains the $σ_8$ discrepancy between the CMB and weak lensing results. Moreover, the self-scattering neutrino behaves as a dark fluid that enhances the size of the Hubble rate $H_0$ to accommodate the local measurement result while satisfying the CMB constraint. For the small-scale structure, the scattering of $\hat{\Omega}$'s through the twin photon exchange generates a self-interacting dark matter (SIDM) model that solves the mass deficit problem from dwarf galaxy to galaxy cluster scales. Furthermore, when varying general choices of the twin photon coupling, bounds from the dwarf galaxy and the cluster merger observations can set an upper limit on the twin electric coupling. ; Comment: 20 pages, 5 figures