what if i could wrote:
I have always been an average runner...but loved doing it...in high school I did low mileage and lots of speed work max 30-35 mpw....and my 5k cross pr ended up being 19:30...mile 5:28 ...2mile 12:12
there were guys who trained the same as me who could run 17:00-17:30 5k ...4:40 mile....10:30 2-mile...
I'm 24. My HS times were about 20 seconds faster than yours (although I stopped seriously running halfway through junior year), and I'd say that our training was almost identical. I had low mileage, no one else running distance to train with, and a coach who considered the 800 a long distance event.
Anyway, I started running again at age 20 to get back in shape and within 7 months had run 18:34 for a 5k. Maybe 30 miles a week, careful diet, and 2 or 3 lifting sessions. Two of the runs were 30 minute progression runs from 8:00 pace down to 5:45 or so, on a treadmill, perhaps a minute at each 1/10mph. That type of training stayed relatively consistent for 2.5 years, although I'm sure the mileage gradually increased to 40-45.
At age 23, I built up to consistent 70 mile weeks and dropped my 5k to 17:30. I also seriously explored 10 milers, 1/2 marathons, and ran a full marathon. The longer races were critical to my psyche because they showed me part of the sport that I hadn't seriously seen before, and trust me there's a huge difference between racing a 1/2 marathon or a marathon and simply trying to complete it. I haven't raced for a while now, but I'm back doing 70 miles per week again for the first time in 8 months following the marathon and it feels easier than it ever has before. I anticipate things only getting better.
I think you'll find a similar progression over the next few years. Your body changes a lot in your early 20's, and it's amazing how hard you can push it and how readily it will respond. It's impossible to tell how much you will improve until you put in the miles, but it's tough for me to imagine you couldn't run at least as fast as my slow ass. Put in the work, be patient, push hard but don't get hurt, and improvement will come. I'd say you're best to increase your mileage maybe 5 or 10 miles per week for every month, if that fast. Consistency is key to building long term aerobic conditioning. Good luck.