The immediate reaction

The Fourth Crusade's sack of Constantinople in 1204 was genuinely traumatic for the Byzantine Greek world — arguably more so, in certain respects, than earlier encounters with Muslim conquerors. The Latin crusaders looted churches (including Hagia Sophia), desecrated altars, melted down sacred objects, and by most accounts behaved with a level of violence and sacrilege that shocked contemporary Byzantine chroniclers like Niketas Choniates, who wrote extensively and bitterly about the event.

The specific comparison to Muslims

The comparison you're asking about isn't a modern historian's invention — it appears in medieval Byzantine sources themselves. Several Greek writers explicitly said that Muslim conquerors (going back to the original Arab conquests, and later Turkish rule) had, in various instances, been more respectful of Christian religious practice, clergy, and property than the Latin crusaders were in 1204. The often-cited (and probably later, though illustrative) sentiment attributed to Byzantine circles — that it would be "better to see the Turkish turban" in the city than the Latin mitre — captures a real strand of feeling, even if that particular phrasing dates from the eve of the 1453 Ottoman conquest rather than 1204 itself.

Why this makes sense in context

A few things fed this:

Muslim rulers under Islamic law generally extended dhimmi status to Christians — a subordinate but protected and defined legal position, with taxation but continued religious practice.

The Fourth Crusade, by contrast, was fellow Christians attacking a Christian capital, which added a layer of betrayal that a Muslim conquest didn't carry.

Theologically, Muslims were an expected "other" — the schism with Rome (formally dated to 1054, though tensions predate that) meant Catholics were a rival within the same faith, and rival co-religionists are often resented more intensely than outsiders precisely because the closeness raises the stakes of the disagreement.

The Fourth Crusade led to nearly 60 years of Latin rule (the "Latin Empire") over Byzantine territory, including attempts to impose Roman ecclesiastical authority — an ongoing grievance, not just a one-time sack.

The longer-term effect

Historians widely credit 1204 as the decisive rupture that hardened the East-West schism into deep popular animosity, not just an elite theological dispute. It fed Orthodox suspicion of Catholic intentions for centuries afterward, and it's frequently cited as a contributing factor in Byzantine reluctance to accept union with Rome even when facing Ottoman conquest in the 15th century — some Byzantines apparently judged continued Orthodox identity under Muslim rule preferable to survival as Catholics.

So: yes, this is a real and well-documented sentiment, though it's worth noting it wasn't universal — some Byzantine elites did pursue union with Rome for political survival, and the picture varies by source and period.

(http://www.autoadmit.com/thread.php?thread_id=5902803&forum_id=2],#50130405)