Simulations with more than $10^{12}$ spins are used to study the motion of a domain wall driven through a three-dimensional random-field Ising magnet (RFIM) by an external field $H$. The interface advances in a series of avalanches whose size diverges at a critical external field $H_c$... Finite-size scaling is applied to determine critical exponents and test scaling relations. Growth is intrinsically anisotropic with the height of an avalanche normal to the interface $\ell_\perp$ scaling as the width along the interface $\ell_\|$ to a power $\chi=0.85 \pm 0.01$. The total interface roughness is consistent with self-affine scaling with a roughness exponent $\zeta \approx \chi$ that is much larger than values found previously for the RFIM and related models that explicitly break orientational symmetry by requiring the interface to be single-valued. Because the RFIM maintains orientational symmetry, the interface develops overhangs that may surround unfavorable regions to create uninvaded bubbles. Overhangs complicate measures of the roughness exponent but decrease in importance with increasing system size. (read more)