What size tire on a 15x4 inch rim ?
Yes to the 165/80/15. That is what is in this photo.
If you do a search on low cost tires they usually are narrower that a well known name (I use IronMan) Web sites usually give specs.
Like a 195/75/15 will have about a 6 to 7 inch tread Just about max for a 4'' rim.Most I find are right at 6''
In the second photo My rim is 5'' and have a 215/75/15 7'' tread IronMan and fits well,no bulge of the side wall.