Late to the party on this, but hopefully I can add something of value.
Based on the initial question, it sounds like you're looking for a Western equivalent of shounen, in which case - as that's essentially the Japanese children's action genre - I'd direct you generally towards children's action shows. I can see both Avatar series have already been thoroughly recommended, which I thoroughly agree with. If you're specifically looking for another show that attempts a Western take on the anime style while maintaining the core shounen action/adventure direction, the original Teen Titans series might be of interest. If the aesthetic style itself isn't as important, I also hear very good things about Steven Universe; Specifically to the context of this conversation, the premise and tone sound like a sort of hybrid of classic Dragon Ball and DBZ, and the crew lets its own anime fandom show through various nods and tributes (I recall seeing a comparison video showing that one particular fight scene was made to match the choreography and composition of a fight in Princess Tutu).
If you're looking for something specifically like Paranoia Agent, I actually have a recommendation that may sound a little strange: Twin Peaks. It goes through some very visible growing pains and has a large stretch that's been all but disavowed by its creator, but with the benefit of the new season it actually follows a very similar arc over a longer stretch: The investigation of a murder slowly reveals supernatural, malicious underpinnings as you explore the interconnected lives of those in proximity to the event, with the narrative shifting hard into heavily symbolic, often dreamlike imagery that makes you question the reality of any given thing you're seeing towards the end. If you're at all familiar with David Lynch's work, you know basically what to expect when all is said and done.
Beyond that, if you're looking for Western shows that are "equivalent" to anime, things get a little trickier. The easy answer would be Firefly given the overt influence of Cowboy Bebop, Trigun, Outlaw Star, and the like (whether or not Whedon admits it), but beyond that you run up against the question of what typifies anime outside of its aesthetic. In my experience, when you remove the differing root cultural norms one of anime's biggest defining elements is innate quirkiness - which even in and of itself might just be a component of Japanese culture, but hey, gotta start somewhere. Between the shared elements with the "slice of life" genre and the mundanely outrageous premises, sitcoms with major fantastical components like Greg the Bunny or the 2002 run of The Tick are the kinds of shows I could just as easily imagine being anime as live action.