Drywall – apply joint compound over drywall primed with PVA

I know this isn't the answer you want to hear, but I don't think priming and painting will hide the tape.

Before you apply any more primer, I'd add another thicker coat of mud. Slather on the mud thick first - it doesn't have to be smooth. Then smooth it out by angling your 12" knife so it's almost parallel with the drywall as you drag it along the surface. You'll have to press fairly firmly in order to avoid air gaps appearing in the thick coat you're smoothing out.

You may have to get a feel for it. If you push too hard and can see the tape again, redo that section until you get it just right.

Wait for the thicker coat to dry, then sand the edges as smooth as you can get them. Very carefully/lightly sand over area covering the tape if you didn't get it quite smooth. You may need to do a second coat along both sides of the thicker coat to make it join the drywall surface smoothly.

There have got to be many ways to minimize the bubbles. These kind of problems you deal with when finishing drywall require a certain amount of skill that, for me, comes only with experience. That said, I have a few ideas:

1. Add a little bit of water. The further along in the finishing process, the thinner the compound should be.
2. Mix well, even if you add no water.
3. "Work" your mud with a knife in your pan (or on your hawk) for a few seconds.
4. It's hard to tell from the pictures, but it looks like your application could use fine tuning. a. Apply the mud to the wall as you have done. b. Remove mud from the knife and, putting pressure on the left side of the knife, go back over application. c. Do the same for the right side. d. With light pressure in the center, go over again. This will help to better feather out our edges.