A technique to obtain high carrier mobility of Sn-doped Ge thin-films (thickness: $\le 20$ nm) on insulators is proposed. In a conventional approach, the grain sizes of Sn-doped poly-Ge films grown by solid-phase crystallization decrease with decreasing thickness. This significantly deteriorates the carrier mobility. To solve this problem, thinning of thick Ge films having large grain sizes is investigated. Moreover, to eliminate the crystal defects, post-annealing is examined. This achieves high carrier mobility $(\sim 200$ cm$^{2} /$Vs) even for thin-films (thickness: 20 nm). A good transistor operation is demonstrated using the thinned Ge films. This technique will be useful to realize advanced thin-film transistors for next-generation electronics.