I think it might be lighting. What lighting are you using? Lighting for a DSLR or Mirrorless camera in your case is SO important. What is your lighting setup? Phones tend to compensate and make poor lighting look better, but cameras don't do that.
Lighting will always improve video! However, any "pro-ish" camera whether DSLR or Mirrorless
can do *everything* your iphone or android phone can do!! The phones are just making a lot of decisions on your behalf to make it easier to use without knowing a lot about how cameras work.
In photography/videography you have a 3-way triangle of trade offs between shutter speed (or framerate), ISO setting and aperture (iris). When you adjust one of those 3 things it will impact the other two things. So its a series of tradeoffs. Phones and point and shoot cameras make these tradeoffs for you with some algorithms guessing what you'd like to accomplish.
1.) For video you really want to target a specific framerate, usually 24fps or 30fps, maybe 60fps for action sports or something. So this decision is made for you. You pick a shutter speed that is about double your framerate. So for 30fps, I set my shutter speed to 1/60th of a second.
2.) Now you have a trade off between the ISO setting and the aperture. Lenses have an iris just like your eyeball. It opens and closes to let in more or less light to your sensor. But also opening and closing the iris in the lens modifies how the lens looks on video (depth of field). So an aperture of 1.4 or 1.8 will get that nice shallow depth of field where your background is more out of focus than the subject of the video. A higher aperture like 5.6 or more will make the whole frame in focus. (Tip: Your phone usually has 1 or 2 fixed apertures that are not adjustable).
Cameras that have modes like "portrait mode" and "landscape mode" are just presets for favour a lower or higher aperture using simpler to understand language. Many YouTubers strive to have a lower aperture number in a studio setting, in order to blur out the background a bit and bring focus to the subject of the video.
3.) So now you have set your camera to 1/60th of a second shutter speed, perhaps the lowest aperture your lens is capable of (more expensive lenses will go lower than the lens that comes with the camera) and you are left with ISO as your last setting.
ISO is like "sensitivity" of the camera sensor. A higher ISO will make the image brighter, and a lower ISO will make it darker.
So if you have poor lighting, you will need a higher ISO setting. The better your lighting the lower an ISO you can get away with.
So why not just always use a higher ISO?? It's a tradeoff!! The higher the ISO the more "noise" you get in the image and the less "crispy" it will look. The lower the ISO the better quality the image will be.
TL;DR; So better lights = lower ISO = better final image quality.
However, if you are on a budget and don't have the equipment right now, just bump up the ISO to 2000 - 4000 range and you should get "cell phone quality" output. But don't judge the camera on that output, because it will look even nicer when you get the lighting required to shoot at an ISO of 100-250 range.
To test out what "nice lighting" will look like, try doing a video on a bright but overcast day with natural light. Those bright overcast days make a giant soft light in the sky, you can drop your ISO down to the low few hundred range while still maintaining a 1/60th shutter speed and see what that video output looks like.
