I trained 6 days a week most weeks and got down to 15:15. Granted, i was 16 and a lot fitter than now! But what gets you faster is teaching your legs to turn over faster and not give up.
Measure a 2k course, or enter some shorter runs. At least once a week do intervals between 3-800m.
BUT, if you’re not used to it, take it very easy to begin with. Do a proper warm up that lifts your heart rate right up.
Something like 4x800m at target 5k race pace, 2 min jog recovery between. Or 6x400m faster.
20min should be absolutely achievable for you. Don’t forget hill sessions. Even a short 4k fast run with as many hills as possible over a set route. Aim to finish the last hill on your knees ?