Chiral edge modes inherent to the topological quantum anomalous Hall (QAH) effect are a pivotal topic of contemporary condensed matter research aiming at future quantum technology and application in spintronics. A large topological gap is vital to protecting against thermal fluctuations and thus enabling a higher operating temperature. From first-principles calculations, we propose ${\mathrm{Al}}_{2}{\mathrm{O}}_{3}$ as an ideal substrate for atomic monolayers consisting of Bi and group-III elements, in which a large-gap quantum spin Hall effect can be realized. Additional half-passivation with nitrogen then suggests a topological phase transition to a large-gap QAH insulator. By effective tight-binding modeling, we demonstrate that Bi-III monolayer/${\mathrm{Al}}_{2}{\mathrm{O}}_{3}$ is dominated by ${p}_{x},{p}_{y}$ orbitals, with subdominant ${p}_{z}$ orbital contributions. The topological phase transition into the QAH is induced by Zeeman splitting, where the off-diagonal spin exchange does not play a significant role. The effective model analysis promises utility far beyond Bi-III monolayer/${\mathrm{Al}}_{2}{\mathrm{O}}_{3}$, as it should generically apply to systems dominated by ${p}_{x},{p}_{y}$ orbitals with a band inversion at $\mathrm{\ensuremath{\Gamma}}$.