Is it normal to have a fever with bronchitis after antibiotics are started?

Depends. Depends on the cause if your bronchitis. If it is bacterial in nature, and you have a fever, it takes time for the antibiotic to work, eliminating the fever. After about 48 hours, the fever should decrease.
With any viral... ...Illness, duration of the fever is the same with and without antibiotics. But if you are feeling worse or having trouble breathing, see your dr again.