Did the city of Nazareth exist in the time of Jesus? The question stems from this video. Additionally, I would like to know any books, articles and other credible sources that deals with this question for further reading*. Thank you for your time.

* History is not my field. I am not experienced in discerning whether a source is credible or not.
It's very easy to look back thousands of years, once time has had plenty of time to change things, and say that there's no evidence for something and so it must be false.  But it's a very different matter when there aren't thousands of years in the way.

The Gospels were written in the first century AD, by people who experienced it personally, to their contemporaries, and they were believed.  Now, miracles are one thing--believing in them takes faith, of course.  But miracles aren't the only things in the Gospels.

If I wrote a supposedly non-fiction account about my own experiences living in the city of Blargsenberg, New Jersey, it wouldn't take long before someone looked at it and said "hey, there's no such city! This is a made-up story!"  But the interesting thing is, there's no evidence of contemporaries rejecting the Gospels on the basis of them contradicting easily verifiable facts on non-miraculous claims.

In the video, the speaker suggests that if there was no such city as Nazareth, the entire Gospel narrative falls apart.  But he does not seem to understand that if there was no such city as Nazareth, it would not have taken people 2000 years to realize that!

Occam's Razor cuts his argument to ribbons.  Now, if no evidence for an ancient town matching the Biblical narrative exists on the present-day site known as Nazareth, a much more plausible explanation is that the present-day site is located in the wrong place.