where's his mistake?

Game 3189210

i thought Bg4 was too premature but was the biggest mistake castling queen side?

his biggest mistake wasn't castling queenside, it was castling queenside when you had a queen right there and a knight a short distance away, and not having anything to defend his king with.

didn't play the opening correctly

Not knowing how to play QGD, 5...b6, 7...e6 and 10...O-O-O lost the game. 3...Bg5 was a bit silly, but it did develop a piece and kept pressure on the kingside. He could have held on had he not ignored the threat from 5. Qb3.

IMO

These moves look less-than-best to me, along with possibly better options:

2...Nf6 (e6, etc.)
3.Nc3 (cxd5)
3...Bg4 (e6, etc.)
4.h3 (Qb3)
5...b6 (Qc8)
7.f4 (e4)
7...e6 (Bg6)
9...Qe7 (Be7)
10...O-O-O (Qd8)
11...Nxg4 (Be8)
12...Bxg4 (Nb8)

Qb3 is almost always good against an early Bc8 development in the QGD. The pattern shows up in a lot of openings - Bishop develops, and opposing Queen moves in the other direction to attack the Knight pawn. Qb3 vs. Bf5/g4, Qg4 vs. Bb4/c5, Qg5 vs. Bc4/b5, and Qb6 vs. Bf4/g5.