A book says using "but" and "even though' in one (compound sentence) is bad grammar. Consider the following sentences: "It is a general consensus that motorists should drive carefully to avoid traffic accidents. But even though bus drivers in my country drive like there's no tomorrow, our country has the least number of cases of traffic accidents." "The teachers argued amongst themselves about the use of 'even though' and 'but' for hours on end. But even though it was established that the two words shouldn't come together in one sentence, I still insisted in using them. Are the two sentences ungrammatical?
"but even though" is fine in itself. Some people advise against starting a sentence with "but", but I personally do not have any objection to this if done sensibly, and your use seems OK to me. There are a couple of other issues:

"It is a general consensus" should probably be "There is..."

"least number of" is wrong, and there seems no need to use "cases". You can say "smallest number of traffic accidents" or "fewest traffic accidents", but this seems incomplete without some scoping statement such as "...anywhere in the world".
Thank you. Thank you.