1. e4 e5 2.nf3 f6 3. nxe5?! fxe5 4. Qh5+ g6 5. Qxe5 Qe7 6. Qxh8 Nf6 and 7.d3 d6 8. Bg5 9. Nbd7 Whites queen is trapped, next move black fianchettos the light square bishop and black queenside castles. d3, Bg5 are not suitable plans. No Bc4 is more solid.
How to continue this game and how to improve my positional play in games as such

1. e4 e5 2.nf3 f6 3. nxe5?! fxe5 4. Qh5+ g6 5. Qxe5 Qe7 6. Qxh8 Nf6 and 7.d3 d6 8. Bg5 9. Nbd7 Whites queen is trapped, next move black fianchettos the light square bishop and black queenside castles. d3, Bg5 are not suitable plans. No Bc4 is more solid.
Just to be clear: I agree that 3. Bc4 and 3. d4 are good moves.
But after Nxe5, black cannot take the knight. Here is an idea to make it somewhat playable for black: https://www.youtube.com/watch?v=1XM9CdS_SOU
Your move. I intend to follow up with Nd5 or h4-h5.

1. e4 e5 2.nf3 f6 3. nxe5?! fxe5 4. Qh5+ g6 5. Qxe5 Qe7 6. Qxh8 Nf6 and 7.d3 d6 8. Bg5 9. Nbd7 Whites queen is trapped, next move black fianchettos the light square bishop and black queenside castles. d3, Bg5 are not suitable plans. No Bc4 is more solid.
Just to be clear: I agree that 3. Bc4 and 3. d4 are good moves.
But after Nxe5, black cannot take the knight. Here is an idea to make it somewhat playable for black: https://www.youtube.com/watch?v=1XM9CdS_SOU
Your move. I intend to follow up with Nd5 or h4-h5.
Elwin 3 ne5 wins instantly
Its computer analysis
Either you win exchange or its checkmate
Try it as an exercise
How to continue the game? Offer a draw. Yes, you COULD sac your bishop for a pawn to open things up very slightly, but why? You'd be down material and your remaining bishop would be bad.