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Recently I played a game with White whose link is:
https://lichess.org/pAoAUp4O/white#42
and reached this position:

I am having a hard time finding why the moves I played next (22. Rxe8+? Rxe8 23. Bb2? ) are mistakes.

In particular, I am interested in knowing why on move 22, it is advisable to play Bb2 (something I find reasonable) and why on the next move, it is a mistake. Why the exchange of rooks makes such a move unsound? I brought the bishop to b2 because it occupies a nice diagonal and it defends my rook on a1 in case of an eventual check on h1 or g1 by the Black queen.

I can only come up with the fact that that with the Bishop on e3 (on move 23), I would be threatening the a7 pawn. However, I am not convinced that is the reason because such a threat can be easily dealt with by playing a6 or simply Bb6.

Besides, is the exchange of rooks (on move 22) wrong because there exists a stronger move or is it just a bad move in itself?

In the diagrammed position you have a material advantage but Black's position is a little bit better because his rooks are connected and he has a safer king position. His immediate threat is RxR, when you have to retake with the king followed by Qg1+ and after you move Kd2 he will play Bb6 and the pressure around your king will become intense. You are paying for your backward development with you queenside rook and bishop playing no part.

Is the exchange of rooks (on move 22) wrong because there exists a
stronger move or is it just a bad move in itself?

It is a bad move in itself because when you play RxR you immediately give control of the e file to your opponent. While you have a rook on e1 you are contesting the open e file.

If instead you play Bb2 you connect the rooks, complete your development and threaten Nxg7 which looks winning. It is a very hard threat to meet. You can meet Re6 with RxR and then carry out your threat. Probably Black has to play Be5. Then you swap bishops and play Ne7+ followed by Nxd5 and Black is just going downhill.

This course of action is much less effective if you start by exchanging rooks and ceding the e file to black. If you try the same tactic with your rook on a1 then black can comfortably play Be5 and retake with the queen after a pretty much forced BxB. With your material advantage you should still win but only after a grovelling defence as it is Black with the initiative making the threats. One of which is just to rush the h pawn down the board.