There is not a camera in the world, whether cell phone or the very best from Nikon or Leica, that will give you a non-blurry photo of moving people if the shutter speed is 1/15. It's not a question of lag between the time that you press the button and when the image is captured. It's just a fact that if the shutter is open for 1/15, that is long enough for any subject movement to result in blur. For any given lighting, the exposure is a result of shutter speed, lens aperture, and ISO. If that results in a shutter speed of 1/15, whether as a result of the camera setting all three parameters automatically or you doing it manually, moving people WILL be blurred, on every single camera in the world.
If you are able to increase ISO (which may increase noise and decrease dynamic range) or increase the aperture enough to get the shutter speed up to 1/60, or better, 1/125, then you have a much better chance at getting non-blurry images of people moving, as long as they're not moving too fast. Slow shutter speeds might also result in blur due to your hand movement while taking the photo.