I added some bleach when I machine washed some white t shirts, & now they have a distinct smell, is that...?

-normal? fine?If you added more than 1/4-1/2 cup, they will smell like bleach. A 1/4 cup should be just enough to make your whites white without the leftover chlorine smell. Using too much can shorten the life of your clothing because bleach is pretty caustic and can break the fibers down quicker than just detergent.
The smell is from the bleach.They probably have a clean smell to them. You can hang them outsie to get rid of the smell or put them in a dryer with a fabric softener sheet.