As several people have pointed out, UTF-16 is not fixed width; you’re thinking of the older, no longer relevant UCS-2, which was replaced by UTF-16 in Unicode 2.0 back in 1996.
The original design of Unicode was to have a single set of fixed with 16 bit characters, but it turns out that that isn’t actually sufficient; it requires compromises like CJK unification that were very controversial, requires ignoring obscure or archaic characters that people still need to use as they may be part of people’s names, used in academic literature, or the like, and provides too little expansion room.
For one example of commonly used characters that are outside of the original 16 bit range (the BMP, basic multilingual plane), the Emoticons (Emoji) block is outside of the BMP.
Even if you did have each codepoint take a fixed amount of space, as in UCS-4 (UTF-32), for most actual text process purposes you have to deal with variable length sequences. Combining characters, for instance, are rendered as a single glyph, but use two codepoints.
There are lots of good sources on this subject already. the UTF-8 Everywhere Manifesto provides some good arguments in favor of UTF-8, including why it should even be used on Windows when the Windows API prefers UTF-16, the UTF-8 FAQ provides some good background information and information on UTF-8 on Unix like systems, Hello World, or Καλημέρα κόσμε, or こんにちは 世界 by Rob Pike and Ken Thompson describing how UTF-8 made it so easy to add Unicode support to Plan 9 incrementally and without having to rewrite the world with different APIs and data structures, the UTF-8 history from Rob Pike describing the original design process, and there are lots of other answers available on StackOverflow, such as this previous answer I wrote, and various discussion forums, such as yet another thread on Reddit, about why UTF-8 is preferable to UTF-16.
As an aside, I’m not linking to my own previous threads because I think that they’re the best sources available, the original sources I linked are better and there are lots of other discussions out there, but just because they contain several arguments I have already made in the past, so I’m trying not to repeat them here.