I haven’t seen it myself, but apparently the show Baywatch Nights fits. The first season featured steamy detective stories. The second season, out of the blue, featured SUPERNATURAL detective stories.
I don’t think it really counts as “getting weird”, given that it was already a series of detective stories about a magician living in Chicago, but The Dresden Files is certainly both weirder and far grander in scope than what one might have predicted from reading just the first book or so.
The Cabin in the Woods starts off with a group of young adults going off to a secluded cabin in the woods, just like the start of any other slasher flick/horror movie. It ends as something else entirely.
Wait… aren’t we tipped off in the opening scene that there is something larger going on? I thought it started in the bunker and introduced those characters before even getting to the kids.
“Seinfeld” started out with the characters navigating fairly mundane everyday situations, but by the last few seasons the situations became more bizarre and contrived and unbelievable.
Tom Waits started out doing jazzy piano ballads, though always much cooler than the average piano balladeer. Then in the late 70s he got bluesier and noisier and more abstract.
The Shadow was, IIRC, a crimefighter with skills and contacts and a double life as wealthy man-about-town Lamont Cranston and – well, that was pretty much it, which was plenty, but wasn’t really the storytelling hook of the age.
But then, when they got Orson Welles to do the voice for a radio show about those adventures, they added in The Power To Cloud Men’s Minds – and, man, that was the missing ingredient: a bona fide super-power, back before Clark Kent made it into print; that made him memorable in a different kind of way, and promptly moved the guy over from “master of disguise with a knack for pistol marksmanship” stories to where he could even take on sci-fi/fantasy problems in between doing impossibly well against mundane challenges – as, if you will, a “super” hero.
Witching and Bitching(original title: Las Brujas de Zugarramurdi) starts out as an action/crime flick that abruptly turns into a horror movie. The whole thing stays funny (intentionally) from start to finish, though.
Comic strips did this all the time, assuming you can call anything in a comic strip conventional. My favorite example is Alley Oop. In 1932 he was a literate caveman who rode a dinosaur (The Flintstones stole from it with both hands.) In 1939 a time machine was introduced and he was switched into present day America and became a science fiction strip. He went to the Moon in the 1940s.
Going in the other direction was Invisible Scarlet O’Neil. Her inventor father exposed her to a ray that made her invisible at will. Like Lois Lane, she stuck her nose into everything and wound up having adventures, including with robots in the 1940s. After a while, though, she lost her invisibility power, a new character named Stainless Steel was introduced and she wound up getting written out of her own strip.
Sort of the opposite direction, but Big Chief Yahoo started as a gag strip where the character had run-ins with The Great Gusto, a W.C. Fields type. Eventually, photographer Steve Roper appeared and took over the strip, making it a straight adventure strip. Later, a new artist added a failed comic character of his own, Mike Nomad. Chief Wahoo and The Great Gusto were never seen after 1947.
Same for Wash Tubbs, which was a gag strip but quickly turned into an adventure strip (generally credited as the first). Soon the character of Captain Easy showed up and took over the strip.
Lost in Space began as a conventional 1960s science fiction show. As such, it was not particularly noteworthy, and I didn’t like it very much (although I did watch it every week, since my hunger for anything SF is considerable). Gradually, surreal and comedy elements began to creep in (usually involving Will Robinson, Dr. Smith, and the Robot). My interest went way up, and the ratings must have gone up, too, since the show pretty much turned into a wild-ass comedy with SF elements.
Smilla’s Sense of Snow starts off as a fairly normal if exotic murder mystery with the hook being that the title character is from Greenland. Later it takes an abrupt left turn into science fiction.
Days of Our Lives started out as a conventional soap opera about a kindly doctor and his children. Then it started adding supervillains, aliens, demonic possession, fetus theft and so many people returning from the dead that even the characters acknowledge that it happens all the time.