Generally, the nasal bridge is not a major contributing factor to problems with nasal breathing.
Deviated septum is the most common cause of nasal congestion. Enlarged turbinates is the second most common cause of difficult breathing difficulty, and usually associated with a deviated septum. Other structural problems with the nose, such as small or weak nostrils, scarring, narrow or collapsed nasal valve may also contribute to nasal obstruction.
Other conditions, such as allergies, medication side effects, chronic sinusitis, irritants, and systemic illnesses may contribute to nasal breathing issues. These non-structural conditions can be treated with medicine and may not require nasal surgery, such as septoplasty or rhinoplasty.
Speak with a rhinoplasty surgeon to help identify and correct such problems. Only after a comprehensive approach will he/she will be able to determine the most appropriate solution for you. Best of luck.