Really? My companions will equip a new bow, but unequip them and use their standard "hunting bow" once combat starts. They also refuse to even equip a bow I give them if I've improved them at a grindstone.
From what I understand, there seems to be a problem if you give them (for example) a good bow
and good arrows (e.g. elven bow+elven arrows). The problem seems less likely to occur if you give a good bow and iron arrows (the worst possible arrows). The bug seems to be regarding the combined damage that is achieved with your bow+arrow types. However, at very high damage (i.e. very good bows+iron arrows) the follower tends to just ignore the bow anyway, and just use the default hunting bow.
Personally, the best bow I have found that works well, is the Forsworn bow. This can also be improved at a grindstone and enchanted, and the followers always seem to use it, along with any type of arrows you give them.