The Catalan minnows are one of four teams yet to taste defeat in the Spanish top flight, but have not beaten Madrid at the Santiago Bernabeu since 1996.
Los Blancos are the only team to win all four of their opening La Liga games and then they beat Marseille on Tuesday to kick off their Champions League campaign with a victory.
Alonso said this was just the beginning and his team was still improving and finding their way.
"We're growing, and the players are getting a sense of purpose," said the Spaniard.
"We're on the right track. And how we'll be in three months' time will be better than what we are now."
Madrid showed grit to beat Marseille after Dani Carvajal was sent off for a headbutt, just days after they played for an hour with ten men against Real Sociedad following Dean Huijsen's red card.
Real Madrid fans can expect rotations in Alonso's lineup against Espanyol, with the coach keen to impress that he wants to keep all players involved.
Alonso benched Vinicius Junior against Marseille but could bring back the Brazilian forward on Saturday.
"There will be moments for everyone - nobody should feel offended by being left on the bench," said Alonso.
One player likely to keep his place is French superstar Kylian Mbappe, who has four La Liga goals after a spectacular start to the season. With a brace of penalties against Marseille, he climbed to 50 goals for the club since arriving in 2024.
Madrid will be without Trent Alexander-Arnold after he suffered a hamstring injury, but Jude Bellingham and Eduardo Camavinga may make their returns from lay-offs.
Both were named on the bench against Marseille but were not brought on as Madrid were forced to battle hard for the three points.
Espanyol should offer another challenge after Manolo Gonzalez's team's solid start to the campaign.
With wins over Atletico Madrid, Osasuna and Mallorca as well as a draw against Real Sociedad, the Pericos find themselves third in the table, level on 10 points with Barcelona and two behind leaders Real.
"Madrid are a very difficult team (to face), especially at their stadium, but last season we beat them in our stadium," said Espanyol forward Javi Puado.
"We're all moving in the same direction, the club, the fans, and the players, and it could be a very good season.
"We're not getting carried away, because it's only been four matches, but if we've managed to do well (so far), it means we can continue doing well."
Elsewhere Barcelona, second, will look to keep the pressure on Madrid when they host Getafe on Sunday night, still at their 6,000-capacity Johan Cruyff stadium as they wait for licenses to reopen their Camp Nou home.
Player to watch: Rafa Mir
Promoted side Elche have started the season in excellent form and target man Rafa Mir has three goals to his name.
With Oviedo, another side that came up from the second division, visiting this weekend the hosts and Mir have a good chance of extending their fine start to the campaign.
Follow your favourite players in the Flashscore app
Key stats
0 - Champions Barca have never lost at home against Getafe in 23 meetings
63 - Alaves have committed the most fouls in the league
345 - Barcelona's midfield maestro Pedri Gonzalez leads the league for completed passes