If you have a sinus infection and bronchitis that has not gone away with antibiotics, you can see an ENT Specialist for an assessment and likely a nasoendoscopy (scope through the nose done in the clinic under local anaesthesia). If there is still infected mucous (mucopus) seen in your nose, a sample of the mucopus has to be taken and sent to the lab for analysis (bacterial culture). This is to determine the type of bacteria responsible for your sinus infection and the antibiotics that will be effective against that bacteria.
You should also be washing your nose with a nasal douche (salt water solution e.g. Neilmed or Flosinus) and apply nasal steroid sprays to your nasal cavities (e.g. Nasonex, Avamys or Nasacort). Most sinus infections do clear with a culture-directed antibiotics. If this fails, you may need a CT scan (speial series of X rays) of the sinuses to assess the severity of your blocked sinuses and the structural abnormalities that caused it.
If you have been refractory to medical treatments and have persistent sinus infection or structural abnormalities on the CT scan, you may need to undergo sinus surgery (functional endoscopic sinus surgery or FESS) to unblock the sinuses. This is a minimally invasive surgery done through the nose under general anaesthesia.
If you do not have a fever or shortness of breath, it is likely that your "bronchitis" is due to postnasal drip from the sinus infection causing accumulation of mucous in your throat region, resuling in phelgm and cough. If your ENT Specialist is concerned about a concurrent lung infection that failed to clear despite treatment of your sinusitis, he or she can refer you to a lung specialist.
Hope this helps and all the best!