The Star Wars: The Clone Wars animated series took place between Attack of the Clones and Revenge of the Sith. The show had to be creative in its storytelling to maintain the overarching continuity of the Star Wars universe. Therefore, though the Jedi in The Clone Wars tried time and time again to defeat Count Dooku, the story couldn't allow that because it doesn't happen until Anakin decapitates him later in Revenge of the Sith. However, Dooku was captured elsewhere in the series, but not by the Republic — he was captured by a band of pirates.
During The Clone Wars Season 1, there is an episode arc that features Anakin and Obi-Wan on a mission to capture Count Dooku, only for him to be captured by a band of drunken pirates. It takes place across Episode 11, "Dooku Captured," and Episode 12, "The Gungan General." The Sith Lord crash lands on the planet Vanqor, the base of operations for an infamous band of pirates. Their leader is Hondo Ohnaka, and his plan was to ransom Dooku for a profit, so he reaches out to the Republic to arrange a trade.
The Emperor sends Anakin and Obi-Wan to collect Dooku from Hondo and his pirates. They are sent with a warning to not underestimate Hondo, but the pair mistakenly does so anyway. They are taken by surprise and are also captured by the pirates, becoming fellow prisoners with Dooku. This forces the trio into an unlikely alliance to make their escape. Even then, they are only able to do so through good luck and the pirates' errors while they are fighting amongst themselves.
Hondo becomes frustrated with the Jedi's consistent escape attempts and he ultimately tortures them. Meanwhile, Jar Jar Binks does his best to save his allies from the pirates, while Turk Falso, one of Hondo's pirates, betrays Hondo and tries to cash in the ransom for Dooku and the Jedi for himself. Eventually, Dooku and the Jedi make their escape thanks to some hijinks by Jar Jar. However, in making their escape, the Jedi lose track of Dooku who kills Turk and steals his ship to escape both the pirates and the Republic.
This adventure does not shine the best light on the Republic army's tactical skills. First, Dooku is captured by a band of pirates, and not the Republic forces. The pirates don't have the connections, resources, or unity of the Republic army, and yet they successfully imprison a Sith Lord. Then, Anakin and Obi-Wan are also captured by those same pirates. This speaks to the hubris that the Republic army maintains — they underestimated the pirates even after being warned not to, and it backfired on them. Finally, they were only able to make their escape thanks to some luck and accidents caused by Jar Jar. The whole mission reflects rather badly on the Republic.