A novel square-fractal-element grid was designed in order to increase the downstream measurement range of fractal grid experiments relative to the largest element of the grid. The grid consists of a series of square fractal elements mounted to a background mesh with spacing $L_0 = 100\, {\rm mm}$. Measurements were performed in the region $3.5 \le x/L_0 \le 48.5$, which represents a significant extension to the $x/L_0 < 20$ of previously reported square fractal grid measurements. For the region $x/L_0 \gtrsim 24$ it was found that a power-law decay region following $\langle {q}^2 \rangle \sim (x - x_0)^m$ exists with decay exponents of $m = -1.39$ and $-1.37$ at $\mathit{Re}_{L_0} = 57\, 000$ and $65\, 000$, respectively. This agrees with decay values previously measured for regular grids ($-1 \gtrsim m \gtrsim -1.4$). The turbulence in the near-grid region, $x/L_0 < 20$, is shown to be inhomogeneous and anisotropic, in apparent contrast with previous fractal grid measurements. Nonetheless, power-law fits to the decay of turbulent kinetic energy in this region result in $m = -2.79$, similar to $m \approx -2.5$ recently reported by Valente & Vassilicos (J. Fluid Mech., vol. 687, 2011, pp. 300–340) for space-filling square fractals. It was also found that $C_\epsilon $ is approximately constant for $x/L_0 \ge 25$, while it grows rapidly for $x/L_0 < 20$. These results reconcile previous fractal-generated turbulence measurements with classical grid turbulence measurements.