At what point did I lose this?

I'm trying my best to understand the consequences of each move, but every opponent I face understands them better. I know I did something wrong here, and more than likely I did multiple things wrong, but I don't have the chess literacy to identify what. Help me please? I'd like to win a game sometime in the near future.

I think I've been placed too high, honestly. I'm rated at about 1000 on That Other Site, I'm facing opponents that are definitely better than me at this! Great brain exercise, at least.

Your move 3 Bd3 is a bit unusual, but 4 c4 is wrong. You should aim for c3 so as to play Bc2 and d4 later on to play for the centre and to bring your Bc1 into play. 3 Bd3 is not wrong, but it is at the root of your later problems. Here is an example of good play

You should recapture 6 Qxf3, as your 6 gxf3 weakens the pawn structure on your king's side. Your king will not be safe there, as is shown in the game.
For that reason 7 O-O is wrong: the king is not safe there. Better 7 Be2 intending 8 d3 to bring Bc1 into play and keeping your king in the centre for now, where it is safer than on g1 since you have captured 6 gxf3.
8 Qa4+ does not help you. Better 8 Bc2 intending d3 to bring Bc1 into play. Your queen is in danger on a4 as will be clear later in the game. Also the queen has given up defence of weak pawn f3 from his Nd4.
9 Na3 does nothing for the centre. Better 9 Nc3 controlling centre squares d5 and e4.
10 b3 is a blunder as he can trap your queen with 10...b5, a direct result of your 8 Qa4+. Here the game is lost. Both sides make more mistakes, but that does not matter.
At this moment both sides have 15 minutes left, while this was 15+15 time control. So you played too fast. You should play the opening slowly and think carefully about the consequences of your moves as you will have to deal with the consequences for the rest of the game. So play slower, think more.

I think you also need to work on your tactics. I don't want to beat a dead horse by sayng your opening mistakes but you need to take your time and reduce your blunder rate. Tactics usually help and try to notice the patterns of tactics.

With that open king (gf3 and 0-0) and all the pieces on the queen's wing you will lose quickly.

The king is your most precious jewel, keep the box closed (f2, g2, h2) and the best bodyguard is the Nf3.

Appreciate the replies, thank you. I don't know what it is about the clock, but having a time limit just sends me into "go go go" mode. I'll try to work on slowing down.

What makes Bd3 an unusual move exactly? What does it prevent me from doing that would be more useful to me later? The other corrections of my moves I understand (not sure why I thought castling with no protection would be a good idea), but I always have trouble developing my bishops in a way that makes sense, and I'm not sure what would be a more standard move.

10 Bd3 is not a bad move, see the grandmaster game above, but the bishop blocks your d-pawn, which makes it difficult to get your Bc1 into play. So the plan with Bd3 is to continue c3, Bc2 and d4 to get Bc1 into play.

c4 is not that bad. Not the best, but still somewhat fine.

The problem with c4 is, that it becomes more difficult to play the d-pawn than with c3.

I prefer c3 and play it often, I just say c4 is fine. I agree c3 is a better and simpler choice.

I'd suggest analyzing your games with Stockfish. Go to Analyze (or analysis?) and switch the Stockfish on. It'll give recommendations and tell if your move is good or a blunder. Generally, analyzing your own games is very educating because you learn what's wrong with the ideas you've had, better than in analyzing someone else's games.