It's a very common misconception. One I even held myself until recently. The more I thought about it, I realized that while I am introverted, I do like being around people. I'm perfectly happy to be with and socialize with people - as long as I don't have to be the center of attention or life of the party. Interestingly it was a 3 month stint doing full-time remote work that made me realize this.
Luckily when socializing with other introverts, they're generally OK with silence and gaps in the conversation, it's not awkward. And when socializing with extroverts they're happy to dominate the conversation and I'm happy to let them and just listen.
Sure, I feel that too sometimes. Sometimes it's because I don't know the other person well so I don't know how they feel about it, so I start to worry that it feels awkward to them and that makes me want to alleviate it for their sake. Often I'm probably just overthinking it.
It can also be tiring when someone dominates a conversation, it really just depends who it is. I have some friends who I could listen to all day, and they'd happily talk all day.
You tapped into something here. I think for some people, like myself, it depends who I am with and my relationship with them. It can be extrovert and they have a constant stream of interesting things which is mostly fun, so it flows naturally... or there are awkward thoughts or moments that lead to "when will this person shut up".
I wouldn't limit and label myself a certain way based on a few situations with a few people. Keep exploring folks!
Luckily when socializing with other introverts, they're generally OK with silence and gaps in the conversation, it's not awkward. And when socializing with extroverts they're happy to dominate the conversation and I'm happy to let them and just listen.