Flight Instructor reported while practicing emergency landings severe turbulence caused a temporary loss of control resulting in terrain alert and descent below 500 ft over a populated area.

Narrative

While performing a training flight; I did an emergency descent/landing scenario with a commercial student. We encountered moderate to severe turbulence during the last stage of the descent; and our airspeed increased to around 130-140mph. This is the edge of our green arc on our indicator. I tried to take controls during this part; as we were nearly over-speeding in turbulence and going towards the ground in a congested area on the sectional. Our scenario landing point was a farm-like field; and despite being in a congested area; I managed to get control of the aircraft around what I believe was 300 feet AGL. We were away from all buildings; towers; and people while we were over this field; but it was still a congested areas. The moderate to severe turbulence; coupled with our high airspeed and a student that is physically stronger than me; caused me to fight for controls during this. Our Garmin device also gave us a warning alert for terrain. After I managed to increase power and climb away from the ground; the flight was uneventful. I believe since this was my third flight of the day; I was physically exhausted; and did not properly set us up in an area for the scenario like I usually do. I apologize for this; it will not happen again.