We have not used antibiotics to treat illness in over 3 years. There are other alternatives. The one I've had the most success with is Super Silver (an oxygenated silver solution). You can order it online. It has no taste (looks and tastes like water) and no, you will not turn blue/silver from use.
That being said, my son had an abscessed area on his gum. We used it successfully to treat and the infected area went away. However, we apparently discontinued use too soon (antibiotics are about 10 days, this, depending on the severity, is 10-30 days -- and the mouth is hard to treat), the infection was still in the mouth and just moved to another area. So this time, we are treating with antibiotics, along with having the abscess drained. Not that the abscess has anything to do with your needs, but I guess I'm saying There is nothing wrong with taking antibiotics when necessary. If she doesn't like the taste, have a pharmacist change the flavor. We had like 10 choices to choose from (ended up with bubble gum). I don't know if there is Publix in your area, but they give the antibiotics FREE so you wouldn't have to pay.
The Super Silver is a great alternative, but like the antibiotics, you have to take it regularly and for an extended period. Bronchitis can get pretty serious especially with a 4 yo. You may want to stick with the antibiotics.