A primary imbalance in Jerome Gambit (1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+) games is White's "extra" pawns vs Black's "extra" piece.
In the following blitz game, the pawns win in the end.
bigskytek - ferdinand122
10 0 blitz, lichess.org, 2023
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+
4...Kxf7 5.Nxe5+ Nxe5 6.d4
6...Bxd4
Black simply returns one of the sacrificed pieces.
The Database has 1,433 games with this move, with White scoring 39%. This is in contrast to the "stronger" (according to Stockfish) for Black, but more complicated, 6...Qh4 - 454 games, White scores 54%.
The lichess.org website is not in agreement. It has 15,833 games with 6...Bxd4, White scores 46%; and 1,956 games with 6...Qh4, White scores 39%.
7.Qxd4 d6 8.Nc3 Nf6 9.O-O Re8 10.Bg5 h6 11.Bxf6 Qxf6 12.Rad1 Kg8
Black has castled-by-hand. White will need to keep his tactics sharp to find advantage.
13.f4 Nc6
Attack or retreat?
Once again, the typical Jerome Gambit case arises: the "normal" move is not as effective as "something else" - in this case 13...Ng4.
14.Qxf6 gxf6 15.Nd5 Kf7
Defending the f-pawn, letting the c-pawn shift for itself.
Upon reflection, Black might have let the f-pawn go with 15...Rd8.
16.Nxc7 Bg4 17.Rxd6
Going a step too far.
17...Red8 18.Rxc6 bxc6 19.Nxa8 Rxa8
The average Jerome Gambit player would be happy here, with 3 pawns for the Bishop.
Of course, the computer would disagree.
Things now unfold as you might expect.
20.h3 Be6 21.f5 Bc4 22.Rd1 Ke7 23.Kf2 a5 24.b3 Bf7 25.Ke3 a4 26.Kf4 Ra5 27.g4 axb3 28.axb3 Rc5 29.c4 Ra5 30.h4 Ra2 31.g5 Rf2+ 32.Ke3
Black's Rook is effective behind enemy lines.
32...Rh2
With 32...Rb2 he might have had better chances: 33.g6 Rxb2+ 34.Kd4 Be8 35.Rg1 Kf8 36.e5 c5+ 37.Kd5 fxe5 38.Kxe5 Re3+ 39.Kf6 Re4 and the progress of White's pawns has been halted.
33.g6 Bg8
Black's Bishop want to be a blockader, but the retreat 33...Be8 was better.
Now the defense unravels.
34.Ra1 Rh3+ 35.Kd4 Rxb3 36.Ra8
Showing the difference between 33...Bg8 and 33...Bf8.
36...Bxc4 37.Kxc4 Rg3 38.Kc5 Rc3+ 39.Kb6 Rg3 40.Kxc6 h5 41.Ra7+ Kf8 42.Kd6
Black resigned