I know you mention wanting to pick fast, that's fine and dandy when you're there, but you've gotta crawl before you walk before you run. If you want to play metal (making an assumption here going off the fast picking thing) go back to the classic rock/early metal roots, and learn to play a lot of those songs. Start slow, the more you do it, the easier it'll get to get fast.
You know, I don't know if it's the route you wanna take, but my experience playing guitar pretty much translated using the pick over to the bass. Dink around on the guitar a bit, and maybe it could help give you some insight into picking on the bass? Either way, guitarists are all about the fast picking, that avenue might reveal something to you that you're missing.
Also, don't be afraid to try different pick gauges. A lot of folks at music stores seem to point you to heavier gauge picks the second you mention you play bass. In my experience, I really prefer the feel of a mid/heavy gauge guitar pick over the super heavy 'bass' picks.
Like a lot of folks in here are saying already, you can play fast with your fingers... You have four of them and on a thumb on your plucking hand, you can play crazy fast. Don't let wanting to play fast be your only motivation for getting frustrated struggling with a pick. I think the best justification for playing with a pick is wanting to make a more crisp kind of sound. Fingers sound smoother, but a pick makes a crisp break. Either way, unless you're playing as a session musician for someone else's album, you don't need to restrict yourself to a certain sound just because past musicians and others tell you you have to. Play the bass how you like, setup your amp how you like, use what effects you like, that's called making your sound. Who knows, you could be the first metal-polka fusion bassist that does nothing but slap through fuzz and an envelope filter. Make your bass sound your way, you don't HAVE to pick if you don't like playing that way.
I agree with the sentiment most of the others in here are saying about your finger strength/fret buzzing issue: you need to get a setup. Go to the guitar store, plug a bass in, and see if you have the same issue. Nobody is going to judge on you, because everyone at the guitar store likes someone who is up and coming who has an interest in music. Play one of their basses and see if you have issues there. It might be your guitar, or your setup, I'd bet it's your setup.
Like the top comment in here, you might want to consider a different teacher if you're just spinning your tires. If I was good enough to teach students (maybe super-noob students) I wouldn't let them struggle without giving them some direction and motivation. Also, consider asking more in depth questions on here, this place is a wealth of knowledge, and everyone in here wants you to be good at bass. Also, use YouTube and the internet, there's an extreme amount of information and tutorials you can get right off the internet. Take advantage of that.
Just don't get discouraged man, playing for a year and a half is nothing compared to the amount of time you're going to play if you don't quit. I've been playing guitar and bass since I was 15, and I'm 29 now. That's almost half my life I've had strings under my fingers. I'm no master, not by any means. Shit, I'd consider myself an intermediate bassist at best. Just try to refocus your motivation for playing: don't have a "I need to git gud" mentality, play bass because you like it. I play bass because I like the feeling of thick strings under my fingers, I like the low sound, I like shaking the floor. If you're playing bass because you like the way getting in a groove makes you feel, or you like the way the sound hits your ear... after enough time, getting good will come naturally.