I'm using the word "suck" loosely here but at least in my case, I notice that I sing MUCH better when I'm either singing one of my songs or if I've made a cover song my own by putting my unique interpretation on it.

I admire people who can belt out cover tunes and sound good on anything they sing but I just can't. Is that normal?

I like having my own voice that works well with my music. I mean, that is what being an artist is all about. I just wish I was more well rounded sometimes and when I find myself not being able to sing certain songs, it's kind of a downer.

Does anyone else go through this?

As long as you can sing YOUR tunes, I would care at all. Unless you want to start a career in a cover band. Then I would care.
You have to find your way. It would be an endless task to be able to sing/play/mix/record/paint/... like everbody else.

It's because you don't care about other people's work. Or maybe you just think too highly of your own work. Who knows..

Keep practicing different styles of cover songs.
The more you do it the better you get.
Even if you're good at singing your own songs now, You'll get even better at your own songs when you practice singing other peoples songs.
Your pallet of colors will expand.
Just like the guitar or piano. The more you practice, the better a musician you become.

I had to stop doing covers many, many years ago when I realized that I just wasn't a good enough player/singer to do them justice. It just felt....wrong.

Got me to start doing my own stuff. That was back in the 70's. I never looked back.

I find that.

I'm not really a vocalist, but I find when I do covers, I can do some songs in more a folk "Iron and winesque" fashion, I can write stuff I can sing well, and its not like I'm lacking range to do other stuff, I just don't feel like I can do those songs justice. I really want to start training and getting more diverse.

There are singers who can be good with covers.... but I always like the original better.
It's a different story when the singer is making the song their own.
Then you can have multiple good versions.

I would go the second way, try to get something out of the song that makes your singing shine. Even better when the whole arrangement is built around your feel.

Don't even try to outshine the quality timbre of the original singer, instead let hear what your voice is good at.

For practice it is good to learn from singers you like by imitating certain qualitys, but when it comes to the artistic goals that is a cover you have to go for your best true own sound and musical style.

Well done you... Johnny Guitar Watson played one note guitar solos that blew other people away cos he had his own style. Patti Smith and Lou Reed wow people with their sincerity and original performances (even though they might not be the best singers on the planet) cos they are great and have their own style. Look at Tom Waits or Anthony Hegarty they do covers but in their own inimitable way.
Sure it's important to learn the fundaments of singing (Breathing/Voice Placement/Rythme) but after that it's down to how you connect to your listeners and being original and authentic is miles better than being able to sing like who'sawhatsit.

People seem shocked when asking, "Can you play (enter any folk tune)?" "I'm not playing covers tonight, just original music."

Or, if they are being an asshat, "Yep, but I ain't gonna."

Most people appreciate that you play original music and enjoy it. But there's always that inbred, braindead butthead who wants Proud Mary...