What has happened is that some really, really, REALLY big web sites like Google and Amazon occupy a teeny, tiny niche where their data storage and retrieval requirements are so different to anyone else's that a new way of storing/retrieving data is called for. I'm sure these guys know what they are doing, they are very good at what they do.
However, then this gets picked up and reported on and distorted into "relational databases aren't up to handling data for the web". Also, readers start to think "hey, if relational databases aren't good enough for Amazon and Google, they aren't good enough for me."
These inferences are both wrong: 99.9% of all databases (including those behind web sites) are not in the same ball park as Amazon and Google - not within several orders of magnitude. For this 99.9%, nothing has changed, relational databases still work just fine.