Is dizziness and faint feeling normal for people with bronchitis?

It can be. If you are sick you may not be drinking enough. Dehydration will definitely make you feel dizzy and faint. Drink, drink drink! Avoid caffeine, however, in what you drink. If aggressive hydration does not help contact your doctor for an exam.
Normal? None of this should be considered normal. Chronic bronchitis must be treated as a serious disorder than needs appropriate management. Dizziness and faint feelings should be considered as a serious disorder that needs appropriate management. Perhaps the treatment of the bronchitis is producing dizziness side effects. Discuss with your doctor the options for management.