Why don't we use data structures more in web applications?

The code feels shoddily written because it is shoddily written. Well written web applications do use these "data structures"; its just that they're called "models" rather than data structures.

Arrays, linked lists, and dictionaries are usually abstracted away by whatever scripting language is in use.

I can't think of common use cases for data structures like heaps/priority queues and graphs. Both come in handy for dependency management and loading assets from disk, like in the case of writing a package manager or a game. But nothing that complex is needed for your typical web app.