I think there is a conceptual thing going on here that needs clarifying (I'll leave the experimental links to others). I presume the question is predicated on the idea that, well, measurements are made within a certain time of each other, which is compared to the distance between the places where the measurements are being made. The concern is that this only gives a bound: if information is transmitted, it happens faster than some velocity which we have now bounded.
However, what one ought to do is consider what special relativity tells you: if two events are space-like separated, there is no notion of temporal ordering. Different observers, travelling at different velocities, can see the events happening in different orders (or simultaneously). So, all you need to know is that the measurement events are space-like separated (i.e. the distance between the events is larger than speed of light $\times$ time between events), and that is enough.
Also, there's a terminology issue. Bell tests do not talk about the transmission of information, but the presence of correlation. The term information would suggest that one party can choose some information to communicate to another party. This cannot happen faster than the speed of light. But the "random decision" made when a measurement is made on an entangled state is somehow resolved everywhere simultaneously, but does not communicate any information.