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You are right that they are in rebuilding phase which is tough for team with such high expectations. Xabi Alonso has to win games and he builds team because if they start losing he could quickly be fired. You are also right that real test will be how they perform against big rivals like Barcelona and Atletico Madrid as those matches will truly show how much team has improved. For now they are doing well enough to give them time they need to get better.
I don't believe he'll be fired immediately. Xabi is a very successful coach, and if his Real Madrid career progresses normally, it could last a few years. Spain is a place he knows well, and he's a coach who's attuned to young players. I still remember what he did at Leverkusen. They didn't lose a single Bundesliga loss for an entire season and reached the UEL final. I have full confidence they'll do the same with Real.
Everyone is adjusting well so maybe this club won't be moving towards any reckless decisions.Because we know that this coach was appointed by Madrid to achieve certain results, which means he now has a little more responsibility. Moreover this lively coach's teaching method is also good because he was not a weak coach before, in fact his best time was in the German league because he was able to guide a club in that league in the right direction through his care. As far as I know Mbappe and Vinicius are the biggest stars currently playing on the Real Madrid field, Especially in this new season, he can commit to remaining undefeated with these players as well because he has previous experience doing this job from a German club.