All my life I pronounced the name of the Chinese capital with the ‘zh’ sound, i.e., Beizhing. It never occurred to me to inquire as to the reasoning we use the French-sounding sound instead of the regular ‘j’ sound. This video by two charming gentlemen you above explains the proper way to pronounce Beijing, and the reason for it. The reason appears to be a very good one – Beijing is a compound word containing two Chinese words – Bei and Jing. ‘Bei’ means north, or northern, and ‘jing’ means capital city. Well golly, that sure makes sense when you put them together, especially considering there is no ‘zh’ sound in Mandarin Chinese! For some reason, Merriam-Webster has both usages listed as acceptable.