Are spellings like privatize and organize Americanisms?
No, not really. British spelling has always recognized the existence of variant spellings using the suffix -ize/-ise. When American spelling was standardized during the 19th century (mainly through the efforts of the great American lexicographer Noah Webster), the consistent use of -ize was one of the conventions that became established. However, since then the -ise spellings have become more popular in Britain (and in other English-speaking countries such as Australia), perhaps partly as a reaction against the American custom. Spellings such as organisation would have struck many older British writers as rather French-looking. The Oxford English Dictionary favoured -ize partly on the linguistic basis that the suffix derives from the Greek suffix -izo and this was also the style of Encyclopaedia Britannica (even before it was American-owned) and formerly of the Times newspaper.
The main advantage of the modern -ise habit : Lazy spellers do not have to remember that there are several important words which cannot properly be spelt with -ize. These include words which are not formed by the addition of the -ize prefix to a stem, but by some other root which happens to end in the same syllable, such as -vise (as in televise), -cise (as in incise) and -prise (as in comprise).
The American system resulted in the creeping of z into some other words where it did not originally belong. Writers of American English should be aware of some spellings that are regarded as incorrect in the UK, notably analyze.