We had known that birds are descended from dinosaurs well before the general public and the majority of paleontologists starting saying "birds are dinosaurs". So simply saying that "we discovered that birds are descended from dinosaurs" is not sufficient to answering your question.

Traditional taxonomy allows for paraphyletic groups, meaning that not all of the descendants of the most recent common ancestor of the group are required to be in that group. So in this case, even though it was known that birds are descended from dinosaurs, they continued to be considered two separate groups, with dinosaurs being a paraphyletic group. Birds were known first, dinosaurs were later discovered and were considered a distinct group, then the link between the the two groups was discovered, but how they were grouped did not immediately change. That birds were not considered to be dinosaurs was a rather arbitrary effect based on how they were discovered and not on any scientific basis.

One book on dinosaurs from 1997 wrote:

In a phylogenetic sense, dinosaurs are not extinct, for birds are theropodan descendants (but see Feduccia 1996 for a dissenting view). For the purposes of this review, however, the term dinosaur connotes what cladists might term "non-avian dinosauromorph." We thus (unrepentantly) use a paraphyletic rather than monophyletic (holophyletic) "Dinosauria." Whatever the scientific merits of the latter, the former is widely understood, and avoids such circumlocutions as "non-avian dinosaur."

A later edition of that same book from 2012 not only uses "non-avian dinosaur" extensively, it also has an entire section on birds.

So why the change? There is a trend in science to prefer cladistic classification, which requires every group to be a clade, meaning that all descendants of the most recent common ancestor of a group are in the group. This effectively means that paraphyletic grouping is being abandoned. So with cladistic taxonomy birds are dinosaurs.

There are other traditionally paraphyletic groups that are still in the process of changing. For example traditionally monkeys were a paraphyletic group, but any clade that includes all monkeys necessarily includes the apes, so in cladistics apes are monkeys. Though, you will still hear many people say 'apes are not monkeys'. Fish was also a paraphyletic group, which included all vertebrates except tetrapods, but of course in cladistics, tetrapods are fish.

As I understand it, it's due to advances in the technology that supports biology and paleontology. When all scientists had to go on were fossils, the bone structure of dinosaurs more closely resembled giant lizards, so thats the conclusion they came to. But recent techniques, including genetic analysis on currently existing species, clearly shows the link between birds and dinosaurs. (And, in fact, alligators are more closely related to birds than they are to lizards.)

I found this article which seems to explain it well:

https://www.scientificamerican.com/article/how-dinosaurs-shrank-and-became-birds/

Science

Jurassic Park in fucking 1993 has a scene where Alan Grant messes with an annoying kid, and he clearly tells the audience that they are similar to God damn birds, you silly goose. You are that kid in the film in real life.

All dinosaurs are reptiles, including birds. The major clade of dinosaurs to which birds belong is called theropods. The other well-known dinosaurs, sauropods (including all the huge quadrupedal herbivores), are totally extinct and have only very distant ancestry with birds and other reptiles.

By the way, crocodilians have been around for 250 million years, so they shared the earth with the huge dinosaurs of old! But they are not dinosaurs themselves.

It's both!

Triassic: Giant Lizards --> Cretaceous: Giant Birds

Science!

I saw some documentary that suggested that they used to chip straight through fossilized feathers and skin to get to the bones because they didn't realise what they were.

The idea is quite old:

Shortly after the 1859 publication of Charles Darwin's On the Origin of Species, the British biologist Thomas Henry Huxley proposed that birds were descendants of dinosaurs. He compared the skeletal structure of Compsognathus, a small theropod dinosaur, and the "first bird" Archaeopteryx lithographica (both of which were found in the Upper Jurassic Bavarian limestone of Solnhofen). He showed that, apart from its hands and feathers, Archaeopteryx was quite similar to Compsognathus.

But having fossil evidence is quite young:

One of the earliest discoveries of possible feather impressions by non-avian dinosaurs is a trace fossil (Fulicopus lyellii) of the 195–199 million year old Portland Formation in the northeastern United States. Gierlinski (1996, 1997, 1998) and Kundrát (2004) have interpreted traces between two footprints in this fossil as feather impressions from the belly of a squatting dilophosaurid. 

https://en.wikipedia.org/wiki/Feathered_dinosaur

Birds are reptiles.

Not a paleontologist, but I think it's a mix of both wrong information being spread back then and also new info being discovered.

I'm pretty sure people knew that birds were dinosaurs for a while, but people just liked the idea that dinosaurs were monstrous lizards. Giant monsters just capture the imagination in a way that giant birds can't.

And then paleontologists started finding fossils that had imprints of feathers still on the body, and it became really hard to ignore that dinosaurs were a lot more bird-like than people would like to believe.

My impression has generally been that once dinosaurs started to be viewed as bird-like, people started to see them as animals rather than as monsters, and that just kinda snowballed into dinosaurs becoming more and more bird-like

It's not X turning into Y when it comes to evolution. There were all kinds of different dinosaurs, that evolved I to all kinds of different niches. Most of those then went extinct due to rocks falling from the sky, or the entirety of India turning into a huge volcano etc.

So it's not that a T-Rex turned into a chicken, it's that T-Rexs lived alongside other dinosaurs that developed feathers and filled more niches etc. those that survived eventually became birds. (Have you ever seen an Emu up close?!? No wonder the Australians lost to them 🤣)

Disclaimer: I have no specific knowledge. Also, remember this happened over (~~billions?~~ hundreds of millions) of years.

Other comments about just not noticing feathers are valid too.

Everyone wants a cool origin story I guess.

Also we moved from "that looks similar-ish" to analysis of fossiles and their evolution + genetic research.

But I, and thats a fact, know very little about that, and come from times when dinosaurs did not wear any fancy feathers.

A large part of the confusion is that dinosaurs did not emerge from birds in the same way that humans did not evolve from Chimpanzees (nor monkeys) - but rather, both groups in those pairs evolved from a common ancestor (but different ones:-).

Birds have feathers and for the most part fly, while alligators not so much. In attempting to simplify, e.g. Avians to "birds", it causes confusion. Alligators also are not warm-blooded as birds are, not do they have beaks, all hallmarks of modern birds, but they do have four legs, long body with a tail, moveable eyelids - and don't they have external ear openings as well? - all hallmarks of modern lizards.

Scientists use precision language like "non-Avian theropod", but those don't map perfectly to common words like "birds", which everyone knows are just government drones anyway:-P.

I'm pretty sure Jurassic Park (the movie) coming out was when it became common knowledge to the general public. A few hardcore dinosaur nerds (and readers of the Jurassic Park novel) were generally aware beforehand, but it was the movie that pushed it into common knowledge. (Source: I was 14 when Jurassic Park (the movie) came out)

I remember in my childhood in the 90s reading it as a possibility. So it was already on people's radar. And as far as I know in the book Jurassic Park they do have feathers. Haven't watched the newer movies, but I heard that in Jurassic World it was explained by the frog DNA that they didn't have feathers.