To get Pure White in world one, you can kill all bosses in the world, up to 1-3, without dying as Physical form, so play always as soul form.
The red dragon can only be killed in 1-2 not in 1-1, each time he passes in the first dragon's path, snipe him from the top of the tower, he'll keep on doing passes on that path over and over again until you kill him.
Current PC Build
CPU - i7 8700K 3.7 GHz (4.7 GHz turbo) 6 cores OC'd to 5.2 GHz with Watercooling (Hydro Series H110i) | MB - Gigabyte Z370 HD3P ATX | Gigabyte GTX 1080ti Gaming OC BLACK 11G (1657 MHz Boost Core / 11010 MHz Memory) | RAM - Corsair DIMM 32GB DDR4, 2400 MHz | PSU - Corsair CX650M (80+ Bronze) 650W | Audio - Asus Essence STX II 7.1 | Monitor - Samsung U28E590D 4K UHD, Freesync, 1 ms, 60 Hz, 28"
