The Latin Empire expands and takes over the Nicean and other post-Roman states, declaring Latin to be the language of state, and making a USA-style declaration of "Give me your poor, sick, and needy, to defend this Kingdom of God" - enacting a fierce settler program, stealing from the Roman Imperial system of taking recruits into the army, teaching them enough Latin to obey orders, and rendering old tongues irrelevant to the soldiers children.
As the Latin Empire claims (and is supported) as the Roman Empire in the East, they push their territorial claims. The settler program provides the core of a large new army of Latins expand in non-Catholic territories, doubled down on by a Pope supporting the idea that fighting for the new Holy Roman Empire in the East is a sure way to have your sins forgiven - provides the manpower, and the ex-Byzantine engineers and ex-military men provide additional generals and equipment.
The new Latin Empire, with expansion and resettlement and privileges for Catholics has a largely Catholic Constantinople, and control over Black Sea trade that boosts their income, backed by a swelling economy from European Immigration.
With support from the rest of the Catholic world, the "Catholic Imperial Period" leads to the slow and steady reconquest of Anatolia, Syria and Egypt, and mass settlement policies cause the cities to be overwhelmed by Catholics.
Whilst Europe suffers depopulation in some areas, those with access to coal and other technologies begin applying greater use of mechanical power as it is cheaper, leading to an proto-industrial revolution, which assists the Latin Empire.
(I didn't use the word, honest guv)