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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 don't understand why there is talk of sacking Xabi Alonso now. This manager has only been in charge of a few matches for Real Madrid, so his skills or tactics cannot be judged by the performance of the club under him in these few matches. It is not that Real Madrid performed well until the last match of last season, but from the match from which Xabi Alonso took charge, Real Madrid started performing badly. Due to player injuries and some other reasons, Real Madrid could not perform very well last season, so we should refrain from blaming the manager unnecessarily. Real Madrid has never appointed such a manager to their club under whose manager the team has performed or can perform poorly.
We can trust this manager because he is a great manager and he has previous experience of managing big teams, so if given time, he will definitely sort out the Real Madrid team again.