I read that she does briefly sit the Iron Throne but due to the treason of Ulf and Hugh, she begins to suspect the other smallfolk dragon riders of betraying her. This leads her to turn on Corlys and Daemon, too. Aegon the Burned retakes Dragonstone while Rhaenyra sits on the Iron Throne, the fleet abandons Rhaenyra because she locks up Corlys, the dragons are mostly slain by rioters and zealots, and Rhaenyra basically loses all her power, so she flees King's Landing but is captured, brought back, and fed to Aegon's dragon.

Aegon II regains the throne for a brief time but the black army led by Cregan Stark takes the city. Aegon is poisoned after cutting off the ear of Aegon III, and Aegon III gets the throne.

edit: but you're right the Aegon III is Rhaenyra's son, not Aegon II's.