The competition between Barcelona and Real Madrid will be even more competitive next season because both teams have made several efforts to bring in a number of players so that the fight for the La Liga title will be even tighter. Real Madrid will try to make several adjustments with their new coach while Barcelona will be much more solid because they are already well connected with the coach. La Liga next season will still be about rivalry and I am sure that both teams will compete for the title because if you look at it, there is no other team that might be able to match them in terms of fighting for the title.
Of the two teams you mentioned, I still think that Real Madrid will still have a little difficulty when they meet Barcelona again. Because the Real Madrid team still has to make adjustments with the tactics of their new coach for next season, while Barcelona already has quite extraordinary abilities plus new players who can help strengthen all lines in the team. So of the two teams, Barcelona might still be superior to Real Madrid because Alonso still has to make improvements to several players who are currently at Real Madrid.
Indeed, every new coach needs to adapt and adjust, but if you look at the matches that Real Madrid have played under Xabi Alanso, it seems like the adaptation process will be faster. Moreover, Xabi is not an ordinary coach, he was successful with Leverkusen, the Real Madrid squad also has many players who have above average skills, and Real Madrid is a big team, so I'm sure the adjustment process won't take long.
Alanso is also the type of coach who likes to try to improve various systems or game schemes to find the right formula. So Real Madrid under Alonso has a clear game concept. Xabi's tactics in several matches so far can be said to be quite good because he is able to overcome various conditions in the match. Madrid's starting line-up formation against RB Salzburg can be proof of this, Xabi Alanso applied the 433 formation but as the match progressed it changed to 3421 with Tchoumeni who withdrew so that CB 2 full backs moved up, so the defense would be stronger, and the flow of the ball would be smoother.
So I think coach Xabi Alonso is cool and I also see Real Madrid next season can compete closely with Barcelona in the fight for the La Liga trophy. Moreover, under the guidance of Xabi Alanso, Real Madrid players are no longer selfish. I guess Elcasico next season will be even more fierce and exciting.