Understanding how electrolyte solvation and nanoconfinement govern sodium-ion intercalation is essential for designing high-power sodium storage electrodes. Here, we use a model series of hydrogen titanates with systematically expanded interlayer spacings to investigate how the degree of electrolyte confinement governs the transition from battery-like toward capacitor-like Na$^+$ storage signatures. In carbonate-based electrolyte, all materials form resistive interphases, exhibit pronounced charge transfer resistance, and show conventional sluggish kinetics. In contrast, diglyme-based electrolyte enables solvent co-intercalation of Na$^+$-diglyme complexes, limits interphase formation, and produces rectangular/capacitor-like CV signatures with strongly reduced interfacial impedance and excellent high-rate performance. Ex situ TEM analysis and operando dilatometry confirm reduced interphase growth and reversible co-intercalation-induced lattice expansion in diglyme. These findings provide direct evidence for a confinement-controlled continuum between Faradaic and capacitive Na$^+$ storage and highlight electrolyte solvation and electrode nanoconfinement co-design as a powerful strategy toward high-power sodium storage.