Goodbye

In the last five years, Hollywood has released

Seeking a Friend for the End of the World (2012, Steve Carell and Keira Knightley)
Melancholia (2011, Kirsten Dunst)
2012 (2009, John Cusack)  although 2009 was almost too late to get that one out
World War Z, Zombieland and what seems like dozens of other zombie movies
This Is The End (2013, Seth Rogan, Jonah Hill, James Franco, Emma Watson and others "as themselves")

And that's just the bigger movies. There's also
The World's End (2013, Simon Pegg)
It's A Disaster (2012, Julia Stiles)
and now
Goodbye World (they can't all be good ones)

Seems like a lot in a row - is it an indication of the Fourth? Or just recency bias?

Most of these are from memory, some from the IMDB list of "end of the world" movies. That keyword is added by users and does not appear to be rigorously monitored. It includes the movies above, along with the Terminator films (yes, but not the primary focus), the Star Trek reboot (well, a world does end), The Tree of Life (perhaps...) and Rebel Without A Cause (huh?).  Still, it's a good starting point.

Keep in mind also that the previous Turnings had 20+ years to build up movies, and this one has had only five - we should check a five-year period within those 20. Perhaps the years 1984-1989 for the Third, 1963-1969 for the Second: That might indicate if it's because A new Turning has arrived, rather than the Fourth.

Looking at the top 10 movies there: The earliest is 1991 (Terminator 2), so there is probably some effect from looking on the INTERNET Movie Database. In the 23 years since, three are from the last 6 years (since 2008, a generally accepted starting point for the Fourth), three from the 5 before that, one from the five before that (that one being Donnie Darko, and those five years including 2001), and the remaining three between 1991 and 1997.

Which isn't very strong evidence - although it seems better if you think the Fourth may have started in 2005. If we look at the next 10 movies in the list, though, all are since 2002 and 6 of them are since 2008 - that makes 9/20 (45%) in the last 6 years. Continuing on with these ratios yields 12/30 (40%), 17/40 (42.5%), 22/50 (44%) ...

It's staying well above the ~25% we'd expect. It really does appear that people are currently attracted to these end-of-the-world scenarios.

(25% being 5 random years out of 20. Using 6 years (2008 to 2013 inclusive) out of 23 (1991 to 2013) gets 26%. Yes, as noted, the keyword isn't strongly defined. However, if it's being used where it doesn't apply that should happen as frequently today as 20 or 50 years ago. And yes, the listing is in order of movie popularity, not applicability of that keyword. Popularity should indicate, though, how much the movie appeals to the overall population.)