Hello, I think you need to look at the standards, which vary by country. For example, you can find the UK standards here in the link provided. As you can see in Table 5, the horizontal radius of curvature varies with speed design.

http://www.standardsforhighways.co.uk/dmrb/vol6/section1/td993.pdf

From a practical perspective I would have thought that a sharp curve is tighter than the design speed whereas a flat curve can accommodate a higher than the design speed.

I agree with Pablo that the starting point is to know what standards apply. Roads are classified with respect to function and constraints. These factors dictate design speed that theoretically takes care of other design elements such as stopping sight distance (SSD). The minimum horizontal radius is a function of design speed and superelevation. This is based on our assumptions on sideways coefficient of friction. (Remember to check that there are no obstructions on the verge in the inside of the curve that limits forward sight distance to less than SSD). Urie's suggestion that the definition of a sharp curve is a substandard curve for the design speed is correct in the context of a road having many curves at or near the minimum. Such a departure from standard must be adequately advised of by means of sharp curve warning signs (see extract of the SADC RTSM attached), to be displayed with an advisory speed information plate.  However, if a road has many substandard (sharp) curves, the design speed should be lowered and the operating speed constrained by means of appropriate speed limits. The next consideration is design consistency (I refer you to research by Rudiger Lamm and NCHRP REPORT 502). On a road that has generally radii substantively higher than the minimum, an isolated curve with minimum radius at the design speed would be ''sharp'' and could surprise the driver, leading to an incident.