We present an experimental study of the velocity circulation in a quasi-two-dimensional turbulent flow. We show that the area rule of circulation around simple loops holds in both the forward cascade enstrophy inertial range (ΩIR) and the inverse cascade energy inertial range (EIR): When the side lengths of a loop are all within the same inertial range, the circulation statistics depend on the loop area alone. It is also found that, for circulation around figure-eight loops, the area rule still holds in EIR but is not applicable in ΩIR. In ΩIR, the circulation is nonintermittent; whereas in EIR, the circulation is bifractal: space filling for moments of the order of 3 and below and a monofractal with a dimension of 1.42 for higher orders. Our results demonstrate, as in a numerical study of 3D turbulence [K. P. Iyer et al., Circulation in High Reynolds Number Isotropic Turbulence is a Bifractal, Phys. Rev. X 9, 041006 (2019).PRXHAE2160-330810.1103/PhysRevX.9.041006], that, in terms of circulation, turbulent flows exhibit a simpler behavior than velocity increments, as the latter are multifractals.