I'm doing C25k too, but I have experience running and am in pretty good shape at the moment. Still, running gets me breathing hard like no other exercise.
I guess it's because I go at 90% maximum exertion rate for the running intervals, though, because the recovery intervals are still long enough that my breathing slows down to normal during them. That would be my first measurement of whether or not I'm going too hard- can my breathing recover during the time set for it.
I also breathe in uneven numbers- three or four footfalls in, two or three footfalls out. Helps prevent stitches for me as I don't breathe on the same side all the time.
I'm going to go against what most people here said and give you the advice my personal trainer gave me: For a short distance like a 5k, being able to carry on a conversation is too little effort. You build your lung strength more efficiently if you run at a level where you can give short, one-sentence answers to a question but nothing more, so don't slow down too much. Ultimately, it's your perceived effort that counts, and that should be at the upper end of your current endurance for your running intervals. I've been doing pretty well with this advice (I'm more hampered by an injury than my lung strength though).
If you want to improve more quickly, cross-train with some HIIT at home. I really don't like strength training at all, but it helps more with my running than anything else I've tried. Plus, the cardio that's sneaked into HIIT will be of use as well.
If you've got allergies, there's just nothing you can do this time of the year- here in Europe, even the most avid runners I know have stopped running outside because everything is pollinating/flowering/blooming all at once this year. I'd suggest running inside on a treadmill, even if it's not as effective, and have another go at outdoor running once the worst has passed. If you don't want to join a gym or don't have/want to/can get a treadmill, you can try running at the time of the day your main allergen is at its lowest- for many people, that's late at night or very, very early in the morning (sunrise early). Also, try running after it's been raining, and adjust your training days to the weather forecast. Less than ideal, but I think it might help with your breathing.
Finally, my most important tip: Have FUN running! Look around you, see all the beautiful things you won't notice if you're in a car or on a bike, smile! Let yourself feel the energy in your body and how far it's come since the beginning of the week's training when you thought you'd NEVER get through this running part. There are so many people looking miserable while running when I'm out on "my" trail, and I always try to give them a nod of encouragement and a smile (although I'm one of the slowest runners out there). I love getting that little acknowledgement and encouragement back, and it makes the next running interval just fly by. Having fun=effort feeling less like effort.
~ I never make misteaks ~
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