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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.
I quite agree with you because he is a vibrant Young coach that can really go a long way in his career, especially now that is at Real Madrid the work will be will be very easy for him since Real Madrid will know from the past used to have quality players. What is needed is a little effort to bring up progress because already the instrument are there and what is just left it's just the instruction on how to fix it. One thing I am confidence about his arrival in La Liga was that he knows in and out of the league because he has came as a player and now returned as a coach, so far the club in particular is not new because he has serve them for many years and I believe he will have gather enough experience on how to lead the team. Honestly, Xabi had done a lot in his career as a coach because what he has done Leverkusen symbolize everything.