Severe turbulence causes large and abrupt changes in altitude and/or attitude and, usually, large variations in indicated airspeed. The airplane may momentarily be out of control.
How much altitude is lost during turbulence?
In my experience, what a passenger my interpret as a long vertical drop in altitude during moderate (as you have described) turbulence would have a resultant altitude loss that would rarely exceed 100 feet. Typically it's less than 100 feet and less than 2 seconds.
Why would a plane fly low the whole flight?
As an example, the Air conditioning packs, that control the pressurization on an aircraft, if one is out, the flight might be restricted to a lower altitude so the system can handle the reduced output of the remaining packs to maintain a suitable pressure onboard.