Is it normal to feel like I can't breathe while trying to ride at a challenging pace?
I think you should always check with a doctor if you have concerns about your health. While you probably can get advice on your health from this forum, this is not the best place to seek specific medical advice that is going to be right for you.
With that said provided you have no issues that can stop you from training here is what worked for me and others to improve:

Try to use a heart rate monitor to train and/or cadence sensor
What you want to do is ride for 30-60 minutes at a constant heart rate and a cadence of 75-90 RPM (Note if you are trying to do strength work going as low as 65 RPM is ok, but note it does put more stress on your knees so listen to your body).
What you want to do is use the gears on your bike to allow you to maintain that cadence and maintain a heart rate that essentially stays flat for the duration of the ride. If you ride in spurts, like you pedal hard and then recover, that will not help as quickly to build your aerobic base up. Your aerobic base will let you ride longer without going above your anaerobic threshold.
Interval or training in Zone 4/5 is something that usually is done less often to help increase speed or functional threshold power (FTP). For example, once you build up a base, can ride around for a while, and know the gear that works best for you to do that you can go to the next harder gear in your rear cassette and try to ride for 30-60 min at 65 - 90 RPM. That will make your heart work slightly harder and force your cardiovascular and skeletal muscles to adapt to the harder work load. Your heart rate will probably slowly keep climbing when you do this and you will probably ride at a slower speed until you fully adapt.

The approach above worked well for me to go from being able to ride only 13 MPH to being able to go 17-20+ MPH while staying under my anaerobic threshold within a year or so.
Here is a boring stretch from one of my rides with metrics where I am doing some of the things outlined above.

Pedaling the whole time.
Managing the gear I am in so that I am able to keep pedaling between 75-90 RPM  and my heart rate in Zone 3/4 for me. Yes in a few small inclines I dip below that cadence, but I usually do strength work between 65 and 75 RPM so it is something I have adjusted to. On a separate note, for some of my personal goals some of my rides focus on Zone 2/3 and Zone 5 depending on what I am trying to work on.
Note your target heart rate will probably be different than mine - Heart Rate Calculator.

Doing some rides like this where you are not trying to push max speed or go up super tough climbs I found has helped and often takes some self discipline to do cause let's face it...going fast and/or climbing hills is fun ; ). You can also take this same approach to a mountain bike on a gravel or off-road trail as well.
Also note there is no way I could do this ride at this effort level 3 months into riding. I still remember having to stop in the middle of a 13 mile ride when I picked up my new bike. However, with doing the things outlined above I eventually got there and you can too.

                
            
Hope that helps.
Addition:
One this I struggle with in the cold is breathing through my nose and often due to the cold breathe better through my mouth. Also as others have mentioned, in sports to get more oxygen into your body you should breathe through your mouth.