The conflict is between Prosper and Bo (and by extension the orphans) who want to remain free or at least not be exploited by the system. Prosper runs away from his aunt and uncle, who only want to adopt his brother Bo, in order to both protect Bo from their inflounces and to try and give him a better life. This conflict extends later to the search by Victor Getz and the attempted exploitation of Barbarossa.
The climax comes when the orphans steal a carousel wing, which is the missing part of a magical carousal that can alter a person's age. When the carousel is fixed, Barbarossa arrives and demands to ride, but is changed permanently when he breaks it. This causes the Conte, the owner of the carousel, to demand Barbarossa's money and shop.
Prosper smiled. "It was just as beautiful as you told us," he said suddenly. "But now it's broken. It was Barbarossa's fault. And I think it will never ever work its magic again."
(Funke, The Thief Lord, Google Books)
The resolution comes as everyone finds their station in life; Prosper and Bo live with Ida; Barbarossa is adopted by Prosper's aunt, but cannot control his thieving tendencies and is sent to boarding school; the other orphans also live with Ida except for Scipio, who has been transformed into an adult and now works for Victor Getz, and Mosca and Riccio, who move into an abandonded building.