Summary: We consider the size spectrum of entrained bubbles under strong free-surface turbulence (SFST). We investigate the entrainment bubble-size spectrum per unit (mean) interface area, \(\mathcal{N}_a^E(r)\), with dimension length\(^{-3}\), and develop a physical/mechanistic model for \(\mathcal{N}_a^E(r)\) through energy arguments. The model obtains two distinct regimes of \(\mathcal{N}_a^E(r)\), separated by bubble-size scale \(r_0\). For bubble radius \(r>r_0\), the effects of gravity \(g\) dominate those of the surface tension force \(\sigma/\rho\), and \(\mathcal{N}_a^E(r)\propto g^{-1} \epsilon^{2/3} r^{-10/3}\), where \(\epsilon\) is the turbulence dissipation rate. For \(r<r_0 \), surface tension is more important and \(\mathcal{N}_a^E(r)\propto (\sigma/\rho)^{-1} \epsilon^{2/3} r^{-4/3}\). From the model, we show that \(r_0\approx r_c=1/2 \sqrt{\sigma/\rho g}\), the capillary length scale, and not the generally assumed Hinze scale \(r_H\). For an air-water interface and Earth gravity, \(r_c\approx 1.5\) mm. The model provides an \(\epsilon -r\) entrainment regime map that identifies a critical dissipation rate \(\epsilon_{cr}\) (constant for given \(g\) and \(\sigma/\rho)\) above which there is appreciable air entrainment, thus separating SFST and weak FST. We confirm the theoretical model and its predictions using two-phase, high-fidelity direct numerical simulations of a canonical FST flow using the conservative volume-of-fluid method: the respective power laws of \(\mathcal{N}_a^E(r)\propto r^{-10/3}\) and \(r^{-4/3}\) for \(r>r_0\) and \(r<r_0\); the value \(r_0=r_c\); the scaling \(\mathcal{N}_a^E(r)\propto \epsilon^{2/3} \); and the predictions of the \(\epsilon -r\) entrainment regime map.