In the manga, not in the tv show

(MANGA SPOILERS) As a child, Himiko Toga killed a bird due to her fascination with blood.

No animal abuse occurs on screen. However, Nezu (a sentient mouse with a Quirk and the principal of the school) was reportedly experimented on in the past, and it caused him to develop a sadistic side (which we see during his fight with Mina and Denki).

Himiko Toga uses a syringe while she fights.

Yes there are tons in the manga.

One episode involves Present Mic getting assaulted by a group of bugs travelling from underground, and they end up covering his entire body. It's not extremely important, so skipping this particular scene is optional.

endeavor exists

At one point, a character does see some sort of apparitions, but they're not really ghosts.

There is a very minor character with snake hair.

Chapter 160 of the manga

Aizawa is pressed to the ground by a strong enemy holding his head down. He suffers injuries and is ultimately fine, but at the time there is the threat of head squashing.

Shigaraki’s sister is killed in the manga

A 5 year old boy named Kota lost his parents when they fought against a particularly strong villain.

The character Magne, a trans women, is killed in a violent way. The villain Overhaul uses his quirk on her, which makes her explode in a bloody way.

The story hasn't actually ended yet, but it seems very unlikely to end poorly

Deidoro Sakaki from the Shie Hassaikai is an alcoholic.

The protagonist regularly breaks his bones due to uncontrollable superstrength, especially early on. This is often shown in fairly graphic detail

Toya Setsuno was cheated on by his fiance.

No, but theres a scene in the first episode where the character Katsuki Bakugou tells the main character Izuku Midoriya to kill himself

In the manga a villain is dropped from the sky and falls too their death in a fight

Several characters’ quirks are dependent on their calorie intake and there are scenes showing them binge eating for that purpose. It’s not a mental health issue for those characters, but watching them eat like that could be triggering for certain people.

One of the main villains has a tic that results in him scratching his neck. The more upset he gets, the more he scratches. There are a couple shots that show red lines, depicting that he's hurt himself. Though not cutting in the usual way, I'd definitely still be wary.

It’s in the overhaul arc, more like steroids then anything else I can Relate it to but that still counts

Bakugo vomits when he gets punched in the stomach by All Might during the Exams arc. It's definitely not depicted as sparkly rainbows that time, both in the anime and manga. Additionally a younger Todoroki and Shigaraki are seen vomiting in flashbacks.

Todoroki was put through very harsh training as a child, and at one point his mother threw boiling water at him. This has affected him greatly, even as a teenager. Also worth noting: in the manga, a major villain (Overhaul) abuses a 6-year old girl while posing as her "father" figure.

The monster creatures known as Nomu have been burned to death but flame hero endeavor

In season 3, Bakugo Katsuki is kidnapped by the League of Villians. They also attempt to kidnap Tokoyami Fumikagi, but are unsuccessful at the last second.

A villain misgenders a trans female character, though her allies do correct the villain in question.

There is a character that uses mind control. Not exactly possession but victims don't have control over themselves

Minoru Mineta regularly sexually harasses his female classmates-he makes inappropriate comments about them, tries to peep on them whenever possible, and has groped them more than once. Unsurprisingly, a LOT of people hate him, and what doesn't help is that it's played for laughs and he hasn't been expelled for his behavior (yet).

One of the villains gets his fingers cut off off screen

During the Yakuza ark one of the villains controls the ceiling, walls and floor of a building. I have personally felt this scene to be quite claustrophobic.

The protagonist is frequently sent to the hospital due to his tendency to break his own bones using his quirk.

Though not in anyway graphic, Todoroki's mom is put in a mental institution by his abusive father after she had a mental breakdown and hurts her son. Todoroki begins visiting her as part of his growth and she seems perfectly alright, if a little lonely. However, Todoroki speaks of this place as if it was a prison and she must be freed from it.

Not quiet a shower, but in true anime tradition there is a hot spring episode which i s basically an out door bath

Quirkless people are heavily discriminated against by most of society.

Smoking but that’s about it

All Might coughs up blood often due to respiratory issues gained from a fight he was in.

y'all PLEASE. body dysmorphia and body horror are NOT the same thing

A character named Eri is abused by her adoptive father.

Episode 17 of season 2 (SPOILER FROM HERE ON) As a nomu is flying away with a captive, Endeavor launches a spear made of fire that strikes the creature directly in the eye. The actual blow is hard to see unless carefully watched due to explosions, but later in the episode after the major battle with Stain is over it reappears. Once it shows back up it has a bleeding empty socket on one side of his head until the creature is gone.

One of the main characters frequently injuries his fingers, which are shown to be various shades of purple and bent out of shape, causing him pain.

A scene where guns are at the forefront is in the beginning of season 3 where the villain Mustard uses one ends up shooting Testsutestsu, but it does minimal damage since Tetsu was hardened.

In the manga there is a group that is compared to the KKK and discriminates against people with mutant quirks.

A character named Minoru Mineta frequently harasses his female classmates is an inappropriate way