Huh? White easily mates in 4 with ... Bg7#
white to move and mate in 5
No, 4 moves is not enough. you missed the clever black defense!
It shows it's a mate in 16 move, when I analyzed the position.....
Poor compiuter or insufficient patience. Chess.com's engine finds the #5 quickly! Make sure it has sufficient local memory available. If slow turn off the other running applications. Also helps to open just 1 position analysis and close all others.

Poor compiuter or insufficient patience. Chess.com's engine finds the #5 quickly! Make sure it has sufficient local memory available. If slow turn off the other running applications. Also helps to open just 1 position analysis and close all others.
Black still has some way to end up with stalemate or extending the game.
Black still has some way to end up with stalemate or extending the game.
The analysis in your screenshot is wrong! As said: "poor computer or insufficient patience". Results in the evaluation column are not definitive. Sometimes they change after a while!

The last post shows black's clever defense after careless white play! Now, how does white defuse black's plan from the start in a 5 moves checkmate sequence?

It shows it's a mate in 16 move, when I analyzed the position.....
You must be using Chess.com's default depth of 20 for Stockfish, which is terribly weak. Go to settings for the Analysis board and change the depth to Unlimited, and Stockfish will find the M5 quickly.
The last post shows black's clever defense after careless white play! Now, how does white defuse black's plan from the start in a 5 moves checkmate sequence?
Obvious once you realize that Black's stalemate attempt requires first moving the Bishop and then the Pawn on the following move. If White misses the move that keeps it at mate in five then Kh2 (once legally playable) still gives mate in six.

"As fast as possible" does not exist for this puzzle and most "professional" puzzles. There is a contract much like in the card game of bridge. The contract for this puzzle only requires of you to checkmate in no more than 5 moves - against the best counter play. Fulfilling the contract means success, falling short means failure. Chess games are similar but the contract is different. Winning is best no matter how long it takes, drawing is neutral and losing gets you nought. That's'why many real game puzzles are weird. They expect you to win a queen but that is insane as game contracts are independent of material balance or game duration. Just get the win, no matter how you do it and how long it takes. Good game puzzles are only those where these factors go hand in hand. You win the queen precisely because it is the only way to win the game and thus fulfill the contract!
Not so easy to solve!
Good luck!