Does "but while" emphasize the meaning BUT?
"But while he said that order had quickly been restored, opposition politicians accused him of minimizing violence."

See a translation

Starting a sentence with "But" generally indicates that the sentence is introducing an idea that somehow contradicts or opposes what has been suggested.

And 'while', in this context, is generally used to acknowledge that something is true before introducing a contradicting / opposing view or piece of information (similar to "even though").

It makes more sense if you read the sentence like this:

"But [even though] he said that order had quickly been restored, opposition politicians accused him of minimizing [the] violence [that had occurred]."

In this context, minimizing basically means ignoring or downplaying.