The only way to go forward while pressing down is to jump. That makes this challenge quite hard.
The hardest part was in 4-2 when I needed to wrong warp. I found a good way at the start to push Mario 8 pxs at the right.
ROM checksum : 8e3630186e35d477231bf8fd50e54cdd
ROM filename : "Super Mario Bros. (World).nes"
Length : 7:03.38 (25460 frames)