TbMnO3 films have been grown under compressive strain on (001)-oriented SrTiO3 crystals. They have an orthorhombic structure and display the (001) orientation. With increasing thickness, the structure evolves from a more symmetric (tetragonal) to a less symmetric (bulk-like orthorhombic) structure, while keeping constant the in-plane compression, thereby leaving the out-of-plane lattice spacing unchanged. The domain microstructure of the films is also revealed, showing an increasing number of orthorhombic domains as the thickness is decreased: we directly observe ferroelastic domains as narrow as 4 nm. The high density of domain walls may explain the induced ferromagnetism observed in the films, while both the decreased anisotropy and the small size of the domains could account for the absence of a ferroelectric spin spiral phase.