Exiting the Sewers
After navigating out of the sewers, Iris informs you that Burgh has returned to the Gym. Clyde also has moved from the entrance, so now it is accessible to be challenged. Talking to Clyde inside gets you a Fresh Water to use, so definitely pick that up.
Castelia City Gym
The Gym puzzle here is that it is a bunch of cocoon elevators, which will eventually take you to Burgh. It is somewhat similar to Volkner's puzzle from Diamond and Pearl. Some of the cocoons have trainers in them ready to...hatch, so if it looks like your path can not go forward, try getting close to it. Another thing to note is that some cocoons have multiple entry ways, so keep that in mind. The fastest path to get through is middle entrance of the first one in the room, right cocoon, right entrance to cocoon, left cocoon and left entrance. You will be taken to an almost tie-dye cocoon room with Burgh inside it.
|1||Clarence||Sewaddle 20||Sewaddle 20||640|
|2||Jack||Combee 20||Swadloon 20||640|
|3||Gary||Sewaddle 19||Combee 19||Dwebble 19||608|
|4||Anders||Dwebble 20||Swadloon 20||640|
|5||Burgh||Swadloon 22||Dwebble 22||Leavanny 24||2,880|
If you have any sort of Fire type (with a Fire type move), Burgh is of no worries to you. The Grass sub-type of Swadloon and Leavanny does not help them in that regard. If you do not, you may be in for quite a ride. Leavanny has a 103 Attack stat, and a 92 Speed stat, making it probably the fastest and hardest-hitting Pokémon thus far. It is also holding a Sitrus Berry, so if its HP gets below 50%, it restores ¼ of its HP. Neither Dewott nor Servine want to go up against this thing. Pignite doesn't really care, as it has swept the entire gym anyway (if you chose Tepig). Poison types like Koffing and Grimer can take some hits from Leavanny, as they resist the hits and are bulky. His other two Pokémon don't put up much challenge, as you have already beaten a handful of Dwebble and Swadloon beforehand.
After defeat, Burgh gives TM 76: Struggle Bug. Struggle Bug is a 30 Power, 100% accurate Special move that lowers Sp. Atk when it hits. If 30 Power tells you anything, it is that Struggle Bug is not a good option for a Bug type move. It gets a 20 power buff in X and Y, but even then it is still useless. Don't use this move, there are better Bug type TMs coming.
With that all said and done, we move on to getting the Bolt Badge and leaving Castelia City!