This began long before the civil wars in Mexico and Spain. Perhaps we could say in both cases that it began with Napoleon, or at any rate, Napoleon in his character as sequel to French Revolutionary thought.
For Napoleon, the state was all. This applied to his followers, too; the Spanish “afrancesados,” that is, francophiles, who represented what we in the US would call liberals or the left, began to attack the Church even in the First (Spanish) Republic. This, in fact, was one of the reasons that the First Republic failed, because it lost the support of the people after its immediate attacks on the Church, which it realized to be its one rival.
However, it came back again in the person of “reformist” cabinet ministers, and the Church and its religious orders were soon stripped of their property and even evicted from Spain during the 1830’s and onwards. This eventually led to resistance from Catholics, expressed during the bloody events of the Carlist Wars at the end of the 19th century.
These currents were reflected in Spanish colonies, too, particularly in Mexico (which, remember, was originally called “Nueva Espana”).
In addition to the native leftist, statist politics, both Mexico and Spain suffered from foreign leftist intrusion, and they seemed to be magnets for radical left-wingers (remember, Trotsky died in Mexico, assassinated by a Mexican Soviet agent).
Interestingly, one of the first things done by the statist forces that you are defending was to attack Catholic education. In Spain, in the 19th century, the government virtually shut down education, because it could not meet the educational needs of the poor who had traditionally been educated by the Church but it was determined not to let the Church continue to teach them. At later dates, in both Spain and Mexico, Catholic school teachers were hauled out of their classrooms and killed by the leftists, and Catholic schools and social inititiaves helping the poor were virtually the first targets of the left.
The state will brook no rivals. This is not to defend violence on the side of the Church and the right, but you have to realize that this was the product of an anti-Church war of over 100 years on the part of the left.
Not defending them at all. Merely pointing out that their presence was the result, as another poster pointed out, of a very different history than America's.
The Inquisition was still active and executing people in both Spain and her colonies in the early 19th century. There was, by law, no religious freedom. The death penalty existed and was enforced for heresy. Not unlike the present penalties for apostasy in Muslim countries.
IOW, the Catholicism against which the leftists of Spain and Mexico rebelled was a part of the state, of the oppressive system against which they fought. Far more oppressive than anything in American history. So, while regrettable, it is not surprising that the reaction against a truly oppressive Church was itself much more violent than anything in US history.