The accurate monitors of the electron beams and γ-ray fluxes are very important for laser-driven photonuclear studies. The flux-weighted average cross sections and average cross sections per equivalent quantum of ²⁷Al($\gamma ,x;x=2pn,pd,{}^{3}He$)²⁴Na reactions were measured above the giant dipole resonance region through activation methods. Laser-driven ultra-intense bremsstrahlung γ-rays, generated by laser wakefield accelerated quasi-monoenergetic electrons, were used in the experiment at the 200 TW laser facility of the Compact Laser Plasma Accelerator Laboratory, Peking University. The results demonstrated good agreement with previous works and were compared with theoretical values given by TALYS 1.9 calculations. The experimental cross sections were confirmed to be 2.4 times higher than TALYS 1.9 with default options. This work proved the feasibility of using the ²⁷Al($\gamma ,x$)²⁴Na reactions as monitors for laser wakefield accelerated electrons and their bremsstrahlung γ-ray fluxes.