Some Anime is niche but there has been memorable anime shows and films that generations of people in the west grew up watching. Dragonball, Robotech, Astro Boy, Speed Racer, Samurai Pizza Cats, Sailor Moon, Pokemon, Gundam Wing and Battle of the Planets for example.
Studio Ghibli movies like Spirited Away and Howl's Moving Castle have also received acclaim in the west. Spirited Away was distributed by Disney and won an academy award. Studio Ghibli movies are regularly aired on film4 in the UK.
Most of the Anime I saw as a kid was on the Syfy network which aired Anime films and movies at night and the early morning from the 90s to around 2011.
Anime can have a negative stigma due to the weebo types just like superheroes can have a negative stigma due to the overzealous nerdy fanboy types.