Apologies if this is over simplifying it, but the answer is just exposure/reps. Now that you have a 5k time, try and run it 1 minute faster. Then do that again and again. Alternatively you can try "sprint" interval training. But still, you just need to build capacity.
Fitness
You just need to do it more.
Personally, aside from frequently/routine, I need to switch it up a lot too. Some days fast and mostly flat. Other days all uphill at a slower pace. Some days, an equal mix, at a moderate pace. Fast days and uphill days seem to make me faster quickest but I need the easy days too.
Also, you don’t say how long ago you finished the c2k things. Was it like a month ago? If so, speed is pretty unrealistic right now, and will get better.
At this level, I would say, run. It's fine to struggle at first, but you should get a bit faster quickly. Objectively, 30 minutes isn't bad at all for a 5km, have a look at your local 5km and tons on people do that kind of time, 20 minutes is already a well trained amateur, and under would be the realm of club runner.
Yeah to be honest I don't think anybody finishes the program with a 30 minute 5K if they genuinely started from a sedentary lifestyle. I was confused by this as well! I think the point is that by the end of the set program, you are ready to run a 5K.
Candidly, a 30 minute 5K is still a big effort for me and I rarely run at that pace. And people think of me as an extremely fit person. I'm just really, really not built for speed.
That's perfectly normal if you're a novice runner and you did not do many cardio activities. When I did C25K my 5k time was around 36 minutes. To improve your time you can look into different training programs, which will often consist of speedwork and increasing overall millage and ensuring your form is good. You're not going to get fast overnight but you can gradually improve your running speed.
There is a lot of training programs online that you can try. If you want guidance from an app I found Run With Hal to be pretty decent.
Am I correct in assuming you weren't very active before completing the C25K program?
If so, you probably just need to work on building up your cardio base. It's fine to go slow. Just keep at it consistently and as your cardio fitness improves, it should get easier to go at a faster pace.
You can also redo the C25K program with jogging during the recovery portion and sprinting during the running portion aka interval training.
What about your height? My wife and I did the C25k together - I'm 6'1", she's 5'6". Took a fair bit longer for her to break 5k in 30 mins than me, but she was working much harder.
2 things to try, a day where you practice runing at higher speeds, like intervals or sprints. Also try doing a "long run" once a week where you do 1.8 to 2x your usual running distance, but at a much lower intensity.
This helps raise your fundamental aerobic capacity so that your usual 5k pace is using a lower percentage of your capacity (which allows you to increase the pace a little).
Besides those things just keep going and stay consistent. Just being consistent in running is a win, and improvement is going to come.