'''Hopping the first bridge and erasing all enemies in the level (Single Player)'''
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At the start of the level when you come out onto the first bridge, turn left and jump off as far as possible. You should land on a ledge below with little to no damage. Now turn around and jump down to the cliff below so as to land in the sort of "V" shaped crevice in the cliff edge. Now proceed along this ledge, hop down to the rocks below and land on a slanted part so as to reduce damage taken. This is illustrated in Sean Oehrle's AoCR run.
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'''Hopping the first bridge and erasing all enemies in the level (Co-operative)'''
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As stated above, get onto the ledge to the left of the first bridge. Now have one coop player jump off and die. With the second player (who is on the lower ledge), jump off the ledge right before the other player respawns, jump to the right so he lands on the V-shaped formation on the cliff. Watch the movie here for a better demonstration of the the whole trick. (Source: HBO Tips & Tricks)