Fluid flows become turbulent beyond a certain velocity. The velocity is almost always with respect to a fixed boundary. However, an observer in a frame of reference travelling with the fluid will also experience turbulence when the velocity of fluid with respect to the boundary exceeds a certain value. Does this indicate that fluid flow is not invariant under a Galilean transformation?
No, because the boundary breaks the symmetry. Turbulence will also occur if a fast moving object (such as a boat or aircraft) moves through stationary fluid. For the observer travelling with the fluid in your example, the boundary is a surface that's moving rapidly with respect to the fluid, and that's what's causing the turbulence.
Invariance with respect to Galilean transformations is an important tool with which one can reason about turbulent fluid flow. Highly turbulent flows are invariant with respect to various scaling transformations as well as to Galilean transformations, and these invariances allow some of the important features of turbulence to be derived.