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To cross the river, you need to complete the “Peallaidh Extermination” quest and obtain Peallaidh’s severed head…

To begin, back in the underground area, backtrack through it to the overworld map of Tokyo. Head to the southwestern portion there - basically backtrack to the tower to Naraku and go as far south, then as far west as possible. Enter the area there to find a bunch of demons.

One of them notes the head of Peallaidh you’re carrying; apparently, these demons were kicked out of Shinobazu Pond by him. Hand it over and they’ll eat the head. In return, the demon horde will opt to act as a bridge for you to cross.