The specs below are for the Warn M12000. (OK, the chart didn't turn out that great, so here is a link to the website it came from:
http://www.warnfactoryoutlet.com/Winches/Heavyweight-Series/Warn-M12000-Self-Recovery-Winch-24v.axd) As you can see, it draws 67 amps without any load. If you are pulling 4000 pounds, it jumps to 210 amps, at 12000 pounds, 440 amps. This massive amount of current is coming from the battery mostly unless you have a special alternator. Your battery has internal resistance which causes internal voltage drop. So, as you draw more current out of the battery, the internal resistance drops a corresponding amount of voltage inside the battery. At the levels of current draw you are getting with a winch, there is a lot of voltage dropped inside the battery. By the way, this voltage drop inside the battery also generates heat internal to the battery. If you draw too much current for too long, you can melt the internals of your battery. If you are going to be using a battery in a winching operation, you might want to check the specs of the battery and the specs of the winch to make sure you don't have a catastrophic battery event at the worst possible time and place.
12V DC PERFORMANCE SPECS
Line Pull Line Speed Motor
Lbs.(Kgs.) FT./min(M/min.) Current
0 30(9.14) 67 amps
2000(910) 14(4.27) 140 amps
4000(1810) 8.62(2.63) 210 amps
6000(2720) 6.92(2.11) 250 amps
8000(3630) 5.80(1.77) 320 amps
10000(4540) 4.39(1.34) 370 amps
12000(5440) 3.86(1.18) 440 amps