Burgh's gym is on the far west street of Castelia City. He has remodeled the gym since the previous game to use cocoons that transport you around the gym.
Battle Your Way To Burgh
Before you make your way to the cocoons of Burgh's gym, talk to Clyde at the entrance to get your bottle of Fresh Water.
Next, go straight up into the cocoon ahead. You will go up one level. Talk to the clown, Harlequin Clarence. He has two level 20 Sewaddles.
Now go into the cocoon to the right of him. You will go up another level. There is a closed cocoon down below. Walk past it and a clown comes out. This is Harlequin Jack. He has a level 20 Combee and a level 20 Swadloon.
After the battle, go into the cocoon that the clown was hiding in. You will battle Harlequin Gary next. He has a level 19 Sewaddle, a level 19 Dwebble, and a level 19 Combee.
Now enter the cocoon from the right side. Fight Harlequin Anders. He has a level 20 Dwebble and a level 20 Swadloon.
Go to the cocoon to the left, and enter it from the left. You will be in Burgh's room.
Burgh has a level 22 Swadloon, a level 22 Dwebble, and a level 24 Leavanny.
When you win, you get the Insect Badge. He also gives you TM76 Struggle Bug. Drop down the hole to the left to go back down to the entrance level.