I've been to all the places you mentioned but don't remember crowds at Peles Castle. I probably just hit that one on a lucky day; it was during the summer of 2015, so quite a long time ago.
With the caveat that there can be some day-to-day variation in visitor flow, even within the same week, I'd say the Alcazar wasn't as crowded as either Casa Batllo or the Vatican Museums. I believe Casa Batllo and the Picasso Museum were the most consistently crowded tourist sites I've ever visited--not that I have visited all the super-popular places in Europe. I strongly suspect both those spots regularly sell out totally. Casa Batllo seemed to be metering people in only as an equivalent number of earlier visitors departed. I remember thinking while I was in Casa Batllo (summer 2016) that it was surprising the local fire marshall allowed such overcrowding, and the Picasso Museum was better only in that no one except me was interested in the area where the artist's ceramics were displayed. Elsewhere it was extremely difficult to get close enough to the wall to read the labels for the paintings.
In March of this year, parts of the Vatican Museums (most definitely including the Sistine Chapel) were very crowded, but other areas (Pinacoteca and Contemporary Religious Art among them) were nearly empty. I can well imagine that conditions are significantly worse at busier times of the year.
I don't remember finding the Alcazar unpleasantly crowded. It's not a tiny place. (I looked for but did not find a size comparison to the Nasrid Palaces at the Alhambra.) The Alcazar definitely wasn't as packed as the Nasrid Palaces were at night, which were worse then than in the daytime since all the nighttime entries are set for the exact same time. I do not claim the Alcazar was a back-door sort of experience, but I've felt no need to warn people about crowding inside, based on my experience there. I visited Seville in April 2019, and part of my visit overlapped with Holy Week. The sidewalks in central Seville were crowded and hotel rates were at least double the usual cost, but it's possible a lot of those folks were visiting Spaniards who didn't go to the Alcazar that week.
I'll be interested to see what other folks say about conditions at the Alcazar.