I have played so many ranked games
Playing a lot of ranked games only makes a difference if your skill is above the ranking you're at.
I know I'm not the best player out there but still shouldn't I have improved to the point where i can carry myself after playing x amount of games?
Pretty sure 50th percentile is somewhere in low/mid silver so not necessarily. Also just because you're playing a bunch of games doesn't mean you're improving. I could play 1000 games but if I make the same mistakes it doesn't mean I'm automatically going to advance in ranking just because I played 1000 games.
to the point where i can carry myself
That's assuming you're playing a carry role, that you are proficient in the carry role, and you've actually gotten to the skill level in which you can act as a better carry than the enemy carry. As a support main I believe I did a lot of work to get myself out of Bronze, but I did do the 50k damage that turned games around, I couldn't have won them without an actual carry. It doesn't matter what my skill level is, if I'm not playing a carry there's no reason to try to force myself to play Annie support every game to 'carry myself out'. Know the place of your role and outplay your opponent, if you don't play an actual carry role let the 15/7 Yas think he carried the game because he probably provided the damage to get you where you are even if he is noob.
is this normal?
It's normal that you can play a whole bunch of games against people of your actual skill level and then not rise, yes. I'm not saying you're necessarily Bronze skill level, but assuming you are Bronze in skill level it'd be normal that after thousands of games you could still be in Bronze if you didn't actually improve.
it seems abnormal to me that I have played so many games in ranked but haven't even advanced outside of bronze
Stop looking at ranked as trying to advance, but as trying to win each individual game, start looking at which champions you do well on statistically in ranked, start looking at what you did wrong when you lost a game. You should be looking to improve your play not raise your elo because improving your play is what improves your elo in the end.