GR says spacetime is curved by mass, right. So what's the basis for explaining the curvature of space (which can be measured, e.g., LIGO) in MOND?
The fact that MOND fits a lot of the data troubled cosmologists, because they know that a General Relativistic theory is needed to explain pretty much the rest of gravity.
TeVeS is an extension to General Relativity that reduces to MOND in the non-relativistic limit. For comparison, General Relativity reduces to Newtonian gravity in the non-relativistic limit. The non-relativistic limit is when speeds and spacetime curvature are small.
Wrong. GR says that gravitation can be modeled as acceleration.