We study field-driven magnetic domain wall dynamics in garnet strips by large-scale three-dimensional micromagnetic simulations. The domain wall propagation velocity as a function of the applied field exhibits a low-field linear part terminated by a sudden velocity drop at a threshold field magnitude, related to the onset of excitations of internal degrees of freedom of the domain wall magnetization. By considering a wide range of strip thicknesses from 30 nm to 1.89μm, we find a nonmonotonic thickness dependence of the threshold field for the onset of this instability, proceeding via nucleation and propagation of Bloch lines within the domain wall. We identify a critical strip thickness above which the velocity drop is due to nucleation of horizontal Bloch lines, while for thinner strips and depending on the boundary conditions employed, either generation of vertical Bloch lines, or close-to-uniform precession of the domain wall internal magnetization takes place. For strips of intermediate thicknesses, the vertical Bloch lines assume a deformed structure due to demagnetizing fields at the strip surfaces, breaking the symmetry between the top and bottom faces of the strip, and resulting in circulating Bloch line dynamics along the perimeter of the domain wall.