You can refrigerate bread but it's not recommended to keep bread in the refrigerator because when you refrigerate bread instead of keeping the bread out on the counter at room temperature you actually shorten the breads shelf life and the bread will go stale and grow mold much faster in the refrigerator than it would on the counter.
Bread should last on the counter at room temperature for at least 3 to 5 days before it goes bad.
If part of the bread has grown mold you can tear off the moldy part of the bread and still safely eat the rest of the bread without risk of getting sick.
In my house our family eats bread within a few days so I never have bread out longer than a couple days so it's never a problem.
But when I tried refrigerating the bread before it made the bread taste bad and it grew mold quicker.
You can place the bread inside a bread box which will help to preserve the bread but you should not refrigerate the bread.