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The western half of Bangkok is called Thonburi, and there's nothing particularly mysterious about why it's quieter. Thonburi was in fact the seat of the Thai kingdom until 1782, when king Rama I decreed that it should be moved to the other side of the Chao Phraya River in order to better defend against the Burmese threatening from the west side. And that's pretty much it: the then-new Royal Palace was built in the east and the modern city grew up around it, eventually including what we think of as the core of modern Bangkok, namely the business and shopping districts of Silom and Sukhumvit. Only in the late 2000s did the Skytrain & MRT train systems even cross the river to Thonburi.
All that said, Thonburi has definitely been undergoing a renaissance lately, centered around the massive Iconsiam mall with its associated hotel and apartment developments and the new Gold Line train connecting them to the Skytrain. And Wat Arun (Temple of Dawn), arguably Bangkok's if not Thailand's most famous temple, has been on the Thonburi side of the river since the 1600s. Check out Thonburi on Wikivoyage for more things to see and do, there's plenty!
Finally, funnily enough, the English name "Bangkok" likely comes from the village of Bang Makok on the Thonburi side, and predates the relocation of the capital. The Thais call Bangkok Krung Thep instead.