Is it true that iron supplementation is safer by mouth than by injection?

Definitely. This is because the transport mechanism in the gut cells is overloaded by too much iron leaving the excess to accumulate in the cell. This is not a problem as cells are turned over in 48 hours with the old cells being shed into the gut and being passed out with faeces. Injectable iron is very toxic but oral iron preparations are extremely safe, especially if they are chelated.