The field of general overviews of the Crusades seems to be dominated by the work of Thomas Asbridge. He's written 2 of the most popular books one can find on them, with a particular emphasis on the First and Third Crusades.
With that in mind, I do want to make sure I'm getting a diversity of opinion and not simply taking the work of one man as gospel (although I'm in no way doubting his expertise). What other writings give general histories of the Crusade's that may provide a different perspective or general conclusion? I want to emphasise this isn't some dogwhistle because I know his work is disliked in certain political circles, I really do just want other reading recommendations.
Apart from Asbridge, the best general history of the crusades is Christopher Tyerman’s ‘God’s War’. Tyerman’s book ‘How to Plan a Crusade’ is an excellent read that illuminates various aspects of Crusade planning and theory
Johnathan Riley-Smith was, until his death, considered the authority on the crusades. His book ‘Crusades: A History’ gives a good general overview, though is much shorter than Asbridge or Tyerman’s works. Riley-Smith wrote most extensively about the First Crusade. I would particularly recommend ‘The First Crusade and the Idea of Crusading’ if you are interested in this period.
Johnathan Phillips has some good works on the Second Crusade and, although I am personally not a huge fan of his work, Thomas Madden offers an interesting perspective on the Fourth Crusade
You might find Steven Runcimans ‘A History of the Crusades’ an interesting read. I would exercise caution, however; Runciman’s work is very dated.
Peter Frankopan presents a new and unique viewpoint on the History of the First Crusade. By placing greater emphasis on Alexius as opposed to Urban II. In his book ‘The First Crusade: The Call From The East’
If you want even more variety you can look at the work of historians who focus on the Arab perspective of the Crusades.