I've written about language reform a couple of times. Here's one focusing on the script and another on the non-adoption of modern Turkish in certain communities, but let's dig a bit deeper here.

It's important to see that the Turkish language reforms of the 1930s were part of a longer process that had started back even before the turn of the century. A number of Ottoman intellectuals - not all of them Turks incidentally - had advocated for a simplification of the Ottoman language, in vocabulary, syntax and orthography. Different individuals focused on different aspects, and the Republican-era reformers continued each of these, changing the alphabet, removing almost all Arabic and Persian grammatical constructs and then stripping out foreign (largely Arabic and Persian) words and replacing them with real or imagined Turkish equivalents.

This was accomplished in a few ways: the national education system; laws; and media (i.e. newspapers and radio) - in other words this was not something that happened passively. There were committees, conferences, publications and, overall, a significant amount of effort that was being put into this. Naturally, Turkish nationalism surrounding the establishment of the republic didn't hurt as a galvanising force! The new national myth-making even led to far-fetched ideas like the Sun Language Theory - which suggested that all world languages derived from Turkic roots - being taught in Turkish schools.

However, the language reformers didn't always succeed. The book by Lewis that I reference in my other answers has a number of examples of words that just simply didn't catch on in the public despite the reformers getting them into newspapers and books. For example, the common Arabic word kalem (pen) persists to this day despite a neologism, yazak, being proposed. You want to see a blank stare? Try using yazak today in Turkey and see how that goes! But for the most part, the process was extraordinarily successful and a speaker today is largely illiterate in Turkish of the 1920s (even if transcribed into Latin letters), let alone the late 1800s, before the first reforms began.

(Someone is going to comment that they can read papers from the 1920s perfectly well, thank you very much, but remember that this is where historians hang out.)