Complete resolution of acute bronchitis can take up to 3 months, so the persistent lung congestion is not worrisome. You should, however, have a chest x-ray to be sure you don't have an unresolved pneumonia. If you do, the cause would need to be determined. It could be useful for you to consult with your primary care doctor or with a lung specialist to determine what makes you "prone to frequent bouts of bronchitis."
The clinical distinction between asthma and bronchitis is not always clear-cut and so you could have had acute, perhaps infectious, bronchitis that resulted in exacerbation of your asthma that you are experiencing now. In that case you could obtain symptom relief with the prescription of anti-asthmatic medicines.
No, you did not start working out too soon. The return to your exercise program, along with weight loss, is evidence of your commitment to good health. Keep it up.
Good luck.