How does radar detect turbulence and wind speed?
The answer you are referring to explains how radar is used to fly through a squall line whilst avoiding the most turbulent areas. 

Conventional weather radar cannot detect wind speed or turbulence directly, it only detects solid and liquid objects above a threshold size.

The most turbulent areas of thunderstorms are also those areas that contain the most water and ice. This precipitation reflects radar waves very well so the most turbulent areas have the strongest reflections and shows up as red / purple on the scope. The area downwind of the core is usually very turbulent as well.

So the weather radar is merely detecting the presence of water. It is the pilot interpreting the picture who is really detecting the turbulence.

Clear air turbulence  (CAT) does not have condensated water in it and is therefore undetectable by conventional radar. There are systems such as LIDAR that can detect CAT but they generally not installed on aircraft.

As @reirab noted in a comment, a Doppler radar can be used to detect the relative speed of the particles from an aircraft. This is not used to reliably detect wind speed, but when there are large variations in wind speed (a.k.a. turbulence) the Doppler shift in the reflections from the particles causes a spectrum spread. This is an indication for turbulence. The traditional S-band airborne weather radars do not have this capability but the newer X-band radar (e.g. this Honeywell radar (PDF)) can detect wet turbulence to distance of up to 40-60 nautical miles using this technique. 



LIDAR (a portmanteau of Light and RADAR) is a technology that uses light instead of radio waves. LIDAR can measure position and relative velocity of aerosols and therefore it can be used to detect turbulence. It has been used successfully to detect wind shears and map wake vortices on final approach from the ground (paperPDF). 

Currently investigation (paperPDF, presentationPDF)is going on into airborne use of LIDAR to detect CAT. One of the difficulty is to detect vertical components of the turbulence. This causes very little Doppler shift but at the same time have the most effect on the aircraft (since the vertical component of turbulence causes changes in angle of attack). 

As far as I know there are currently no commercial applications of LIDAR for airborne turbulence detection.