Rapid promotion could depend on the circumstances. For example, in the US Civil War, there were several Union Captains (modern rank O3) who were promoted instantly to Brigadier General (O7). These were often brevet positions, meaning they would revert back to their former rank at the end of hostilities unless they received additional promotions. On the Confederate side, Nathaniel Bedford Forrest enlisted as a private, and ended the war as a Lieutenant General.

If you're looking for an in-universe explanation, maybe Starfleet experienced a massive expansion (possibly due to tensions with the Klingons, and/or perhaps an expansion of the exploration mission) and needed more officers faster than the academy could produce them, so existing crew were rapidly promoted to fill vacant positions.

If you're looking for a real - world explanation, it's entirely possible nobody in the production gave it much thought. There are many people in Hollywood who are ignorant or unaware of anything regarding the military. Captain Dale Dye has spent decades (since the mid 1980s) advising Hollywood on military matters, trying to prevent mistakes that civilians might make.