Topological insulator spin-polarized surface states are attractive for spintronic applications, in particular for spin-charge current interconversion, where extremely high conversion efficiencies are predicted. However, the contribution of topologically trivial bulk states is often disregarded although it may play a crucial role in the experimental results and extracted conversion efficiencies. The presence of bulk states at the Fermi level can be avoided by increasing the gap using the confinement effect appearing as the film thickness is reduced. We address this topic by growing ${\mathrm{Bi}}_{1\ensuremath{-}\mathrm{x}}{\mathrm{Sb}}_{x}$ thin films (2.5--15 nm) by molecular beam epitaxy on InSb, ${\mathrm{BaF}}_{2}$, and Si substrates. The surface electronic band structure is studied by angle-resolved photoemission spectroscopy. Two ${\mathrm{Bi}}_{1\ensuremath{-}\mathrm{x}}{\mathrm{Sb}}_{x}$ surface states are observed in the gap for several Sb concentrations and thicknesses, across the topological insulator phase, scanning $x$ between 7% and 30%. Tight-binding calculations of the surface states are in good agreement with the experiments, revealing their polarized nature. Surface states are still present at the \ensuremath{\Gamma} point for the thinnest films (2.5 nm), suggesting highly confined polarized states at the surface.