To reduce 7 minutes on a 5km race is totally doable, but you need consistency. Three workouts per week is too little if you're not making the most of them - the structure mentioned above is on the right track, but I'll tell you from experience that there's one detail most people ignore: active recovery training on non-running days. A lot of people think complete rest is ideal, but walking for about 20-30 minutes at a comfortable pace (seriously, very easy) on those other days helps a lot with improving circulation and adapting your body to the effort without overloading it.
As for the training itself, you don't need anything complicated. Break it down like this: one speed day (like 8-10 reps of 400m fast, with 1-2 minutes rest between them), one tempo day (continuous running for 20-30 minutes at a pace where you can say short sentences but not paragraphs), and one longer day on the weekend (like 6-8km nice and easy). In the first 4-6 weeks you'll already see improvement because you're basically teaching your body to be efficient.
One more thing: get a phone with GPS or use an app like Strava to track every workout. Knowing exactly what pace you're running at (not "I think it's fast") makes all the difference. At first it's frustrating because it looks slow on paper, but the numbers don't lie - when you see the progress week by week, it gets a lot easier to keep the motivation going. Good luck out there.