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First of all, what do you mean by "improve the shutter speed"? Do you mean allow a faster shutter speed?
Flash essentially has no effect on shutter speed in and of itself, and shutter speed has no effect on flash. Flash adds a vey fast burst of light, much faster than any standard shutter. What the extra light allows is lower ISO settings and/or smaller apertures.
If you were in a totally dark room and lit a subject with only flash, if wouldn't matter if the shutter was open for 1/250 second or for one hour: the subject would be lit with the same 1/1000-second (or whatever) burst of flash.
In reality, we rarely shoot in total darkness, so the shutter speed does matter because slower shutters will allow more of the ambient light to record in addition to the flash.
Going in the other direction, too fast of a shutter speed can cause problems if it is faster than the camera's maximum "flash sync speed," because at higher speeds the shutter is never fully open. Instead a moving slit of partially-open shutter curtain moves across the sensor or film, exposing only part of it to the ambient light at any moment. The flash pulse will only light up part of the scene in that case, the part under the slit at the moment the flash fires. Everything else will be lit only by the ambient, and so will be under exposed.
Now, back to the question: let's say you're shooting a wedding reception, and you want a picture of the bridal couple dancing. The room is fairly dark, and if you shoot at ISO 800 and f/5.6 the ambient light requires 1/60 shutter speed. If you add flash, there's enough light to shoot at 1/125, which reduces motion blur a bit better than 1/60.
When you say "What I am seeing is adding a flash does not improve the shutter speed" can you give more detail on what you mean? What are you shooting, under what conditions are you shooting, and what settings are you using?