The London Dungeon is a popular tourist attraction recreating macabre historical events through live actors, rides, and special effects. It is not in West London.
The South Bank is a cultural corridor including the National Theatre, BFI Southbank, Royal Festival Hall, and Tate Modern (further east). The London Dungeon moved here in 2013 from its original Tooley Street location (near London Bridge, also South Bank, not West London).
West London typically refers to areas west of the City: Kensington, Chelsea, Hammersmith, Fulham, Chiswick, Ealing, Notting Hill. The London Dungeon is firmly in Central/South London.
If you're visiting from West London, take the Circle/District line to Westminster (15–20 min) or the Tube to Waterloo.