Autofocus speed and shutter lag are related, but not the same. When you depress the shutter, the camera has to both lock focus and determine the appropriate exposure, then capture an image.
Some of the newer small cameras have a phase-detection focus system that can begin to approach SLR levels of speed and accuracy. Most don't.
Your exposure and AF settings will determine how laggy your shutter is due to exposure calculation. If it is set to spot, and that spot is the AF point, you will have much lag.