As a Madrid fan, I wouldn't overestimate whether Madrid could improve after recruiting several new players and bringing in a top-tier coach. I'm not underestimating Madrid or Alonso, but I simply hope Madrid can pursue more prestigious titles like the Champions League. Even if Madrid fails to win La Liga, at least Alonso's team managed to lift the Champions League trophy this season, which is a more valuable trophy. Still, Madrid's improved performance will make them better prepared to compete with Barcelona, and La Liga will become more competitive.
I think Real Madrid should focus more on chasing their domestic title first, then they can pursue titles outside of La Liga. Barcelona already won La Liga last season, so now is the time for Real Madrid to win that title again. once they can secure the domestic title, they can focus on chasing titles like the Champions League, etc. because right now the race for the European title is fiercer, and if Real Madrid can't prioritize what they need to get, then last season's results will repeat themselves where they failed to get any title.