I have the same apprehension, but anime is a medium. There's a lot of disgusting books and Western movies around, and we don't shame people for liking books and Western movies.
The problem with anime is that some of the most popular ones suffer from the issues described in the meme. Take death note for example, immensely popular. But the female characters are mere props with one massively problematic female character's infatuation with the lead driving half the show. It's just unwatchable if you don't like that kind of thing.
Here's some genre keywords for a start:
Adult Cast
Seinen/Josei
Chances are if those shows involve underage sexualization/unhealthy relationships, it doesn't endorse them.
Edit: Also look for anime with kids as target audiences, they should have none of those things. On further introspection, that sums up to "look for shows that ain't shounen or shoujo" lol.
And if you want some fan service that fits these requirements and is still wholesome I give you Goldenboy.
Why am I not seeing One Piece in here? It sexualizes nami and Robin but they are both over 18. There aren't really relationships at all in the show unless you count boa hancock but that's pretty tame and not a relationship
Vivi is 16 before the time skip and the guys try to spy on her in the bath. Rebecca is 16 and wears a chainmail bikini for an entire arc. Shirahoshi is 16 and wears a bikini top while a creepy guy stalks her for an entire arc.
