Notes

"Anime is just for kids" or "It's just a cartoon" People tend to often look at something animated or cartoonish and assume it's kid-friendly, and Cartoon network's dub of Naruto, or Nickleodeon's Avatar the Last Airbender, or Ledged of Kora didn't help, especially with both Avatar and Kora both being outsourced to Japan for quality animation for their show.
Pokémon, My neighbor Totoro, Ponyo, and Kiki's delivery service, are also for all ages. Naruto teaches some great lessons that everyone could stand to learn from; No dream is too big, to not get stuck in the past, and it's never too late for redemption. A lot of anime as a whole have some fantastic lessons q few examples; Fairy Tails' (Fantasy/adventure) main theme is that family, doesn't necessarily mean 'bound by blood, but deep connection with others' unfortunately Fairy Tail is pretty adult with visual and language, but a good show over all. My hero academia (Adventure/fantasy) teaches many lessons including; your actions have consequences, rivals can strengthen you, and to not let others or odds defy your goals.

Regarding assumption, I'm using the Magical Girl shtick, magical girl anime are often made and geared to little girls, but the actual characters are girls who posses superpowers and have a very cutesy, girly look, with the main character being pink usually for the appeal, accompanied by a group of other magical girls with similar power. These magical girls are tasked with protecting the world from the "weekly monster" or alien.

A good example of this is Tokyo Mew Mew. Tokyo Mew Mew is about these 5 girls who DNA gets infused with that on an endangered species, giving theme some kind of nod to the animal, for example; cat ears and tail, with some kind of power and natural abilities these animals possess. Mew Mews are their superhero brand in show, they fight against aliens who claim they want to 'reclaim' the earth for themselves. Tokyo Mew Mew came out early 2000's and was rebooted this year.

Now that we have that out of the way, lets move onto the added genre that we have in anime for the original purpose to shoot down the "it's for kids" thought, Horror. My example and reference is the show that has been voted #17 for top horror anime before, Madoka Magica.
"The story follows a group of middle schoolers, led by the main protagonist Madoka Kaname, who makes supernatural contracts* to become magical girls. In battling surreal enemies known as 'witches'*, they learn of the anguish and peril associated with their new roles." These contracts are formed based off a single wish, that can go against the laws of nature, and this in turns make the user's soul into a physical form, called a 'soul gem' theses gems can become corrupted, turning our characters into a 'witch' for the others to take down. Aside from the dark base, there is also very strong language, mainly at the tail end of the anime, along with disturbing visuals, such as; Tomoe Mami's head getting bitten off in one of the first episodes, and Sayaka Miki gracefully dancing with her spear, before plunging it into her chest, killing herself.
With all that being said, I don't believe all anime is just for kids, or a cartoon. I am a huge anime fan and watch it regular and there truly is something for everyone, I've even managed to drag my younger sister and mom into it