♡ 39 ( +1 | -1 ) gunnar, Black could not play Nxe5 - There is no black knight on c6 yet, in case you haven't noticed ....
And to answer the original question: Yes, 4.Nxe5 is the best move. Black is getting no edge in development. In fact , it's white who is getting the better development and the center after playing d2-d4 soon. Not to mention the extra pawn of course ...
Therefor, black's 3rd move was an error, and 3...Nc6 was better.
♡ 96 ( +1 | -1 ) 4. Nxe5 gives White a lead in development; it doesn't put him behind. White will also have better control over the center since he has an e-pawn, Black has no pawns in the center. So, 3... Bc5 is inaccurate; 3... Nf6 is more common and leads to an even position.
A common continuation is 4... Nxe5 5. d4 Bd6 (5... Bxd4 6. Qxd4 speeds up White's development even further while Black gives up one of his bishops in a wide open position; most other moves amount to a gambit of a Black pawn) 5. dxe5 Bxe5. White's development lead is going to show as he can quickly deploy his remaining minor pieces while Black still has to move his d-pawn to bring out his other bishop. Black has the option of playing ...Bxc3+, ruining White's pawn structure, but this drops Black even further behind in development, gives up a bishop in an open position, and gives White even more open lines for his pieces.