First, asking how "accurate" it was is to somewhat misunderstand history as a discipline. In history there are certain types of claims -- names, dates, economic data, population data -- that are purely factual. Another important part of history as a discipline is interpretive and consists of discussing not just what happened, but why certain things happened and how people understood events. It is within this framework of history as interpretation that we should look at Jackson's central and seminal claim:

The existence of an area of free land, its continuous recession, and the advance of American settlement westward explain American development.

The first thing we should acknowledge is that this claim was a landmark in the understanding of United States history, the sort of statement that was boldly original when first advanced and of great explanatory power. It marks out a distinctly American type of self-awareness reflected in our popular culture that is quite unlike that of Europe.

Despite its great explanatory power, it tells only part of the story. First, the notion of the "frontier" does not take into account Native American viewpoints, in which European invaders were conquering and destroying their lands. Next, Turner's triumphalist take on the notion of the frontier and rugged individualism is not one shared by more recent historians. Also, his account tends to focus mainly on men rather than women. In terms of Turner's correlation between the west and democracy, there are concerns that he considered his society to be democratic in spite of the fact that it was somewhat less equal, especially with regards to race and gender, than his rather idealistic conceptions suggested.

On the other hand, his work marked a major shift in the study of American history, from being concerned with wars and presidents to emphasizing a broader type of social and economic history. This is still of lasting influence. Although historians have developed more complex and nuanced responses to the notion of the frontier over the past century, people still read and respond to his work and it is of major and lasting significance. Isaac Newton famously said " If I have seen further it is by standing on the shoulders of giants." Turner remains one of the giants of American historical scholarship, even if the historians standing on his shoulders see further.

It is very difficult to answer a question like this in an objective way because much of Turner's thesis had to do with his interpretation of what impact the frontier had on the American character.

Many people today disagree with Turner's thesis.  It seems old-fashioned to them with its emphasis on the idea that the settling of the West was a positive process that improved the country.  Modern historians are more likely to want to look at the ways in which the frontier impacted Native Americans and women and Hispanics as a way of making our view of the frontier more nuanced.  It is hard to prove that modern ways of looking at the frontier are better or worse than Turner's, just that they are different and show the impact of changes in the way we think about things like race relations.

One thing that we might think about is whether we have actually lost the characteristics that Turner gave the frontier credit for.  Have we really become less democratic and less self-reliant?  If you would say we have (and again, this is opinion) then you might conclude that Turner was right.