You need to get out more. Bows and crossbows have no kickback because all of the "push" the bolt or arrow puts on the string is taken up by the fact that it's far far lighter. Force = Mass * Acceleration.
The reason a GUN has a kick BACK is because you're making a small explosion inside the firing chamber which pushes in all directions: fires the bullet out one end and kicked the butt of the gun into your shoulder.
Taking a bow+arrow as a closed system, when you launch the arrow out one end, the bow will go in the other direction. Mass and velocity determine the
degree of kickback, but if you tossed a pebble off of a space station, the station would experience a small recoil.
Recoil is large for guns because of the large velocity of the bullet, regardless of how that velocity is achieved. If you had a gun that fired by using a very strong rubber band that got the same bullet velocity as gunpowder gets you, it'd have the same net kickback as a traditional gun. You can reduce the instantaneous kickback (i.e. the amount of force you receive at any given time) by accelerating the projectile gradually; this is where the explosion matters, because the bullet gets all of its velocity in a minute fraction of a second, so all of the recoil is generated in the same amount of time. The launch of an arrow, on the flip side, is comparatively slower. Your body still absorbs a large amount of force (arrows could easily be heavier than bullets, especially handgun bullets, but they tend to have lower velocities), but if it's over a longer period of time, you don't notice so much.