Travelling at faster than mach 1, how sharp could Concorde turn to avoid injury to passengers?
All aircraft experience the same G force per degree of bank angle, so a 60 degree turn in a Cessna produces 2G just like a 60 degree turn in Concorde would produce 2G. 

The difference is that a Cessna doing 100 knots could circle in 0.2 miles by using a 60 degree bank angle.

Concorde doing 1000 knots and 60 degrees would take up 19.5 miles to do the same thing.

In both cases the passengers would experience 2G which even so is pretty severe if you are a passenger and aren't used to it. It would get your attention let me put it that way.

I'm not an airline pilot. However I would think that the risk with higher speeds is in angle of attack or angle of attack combined with bank angles. At higher speeds its much easier to produce a whole load of G by pulling or pushing the controls than it is in your Cessna, and even that can be pretty severe. 

So while you might not worry as much about bank angles you would be very careful not to pitch up or down too far especially at higher speeds because you could overstress the aircraft real easy.

Hopefully a pilot who knows Concorde and / or airliner systems can improve on this answer.