An "objective" assessment of the Jerome Gambit is that Black should win.
A practical assessment of the Jerome Gambit played at the club level, however, is that it can be remarkably effective for White.
The successful path of the first player, it must be admitted, involves passing through initial dangers.
In the following game, Bill Wall is accompanied in his journey by four passed pawns, and they chaperone him successfully.
Wall, Bill - Twister
internet, 2024
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+
4...Kxf7 5.Nxe5+ Nxe5 6.Qh5+ Ng6
7.Qxc5 d6 8.Qd5+ Ke8 9.O-O
Black has the typical Jerome Gambit piece-for-two pawns advantage.
White's castling makes sense, before he is distracted by threats to his Queen.
Bill has also played 9.d3 c6 10.Qb3 as in Wall,B - Seven11, Chess.com, 2008 (1-0, 51); and
9.d4 N8e7 10.Qb3 as in Wall,B - Am53, PlayChess.com 2013 (1-0, 14) and Wall,B - Guest7665109, PlayChess.com, 2019 (1-0, 22).
9...Nf4 10.Qb5+
Bill has raised this question in similar positions in other games - Block the check with the Bishop and lose the b-pawn? He now suggests 10.Qa5.
10...c6 11.Qc4
A slip, unusual for Bill, as Black now has the murderous 11...Be6. Instead, the second player plays an even more dangerous move.
11...Qg5 12.g3 Bh3
This move is okay, but 12...Qh5 (or 12...Be6) was stronger.
13.Re1 Qg4 14.e5
Opening the e-file against the enemy King bids to keep White in the game.
The Queens can now come off the board with 14...Qf3 15.Qe4 Qxe4 16.Rxe4 Ng6 17.exd6+ Kd7 and both Kings looks safe.
14...d5 15.Qxf4 Qxf4 16.gxf4
Stockfish 16.1 evaluates White as being about a pawn better, but that extra one is passed and protected.