As an American, this is a tough one, because I have a lot of respect for what that man did. According to Navy records, he earned a silver star and three bronze stars. I know one silver star veteran personally, and he's frankly a better man than I'll ever be. There have been books (plural) written about the action where he earned it. They don't give those stars out in cereal boxes.
That being said, its a matter of record that in his book he claimed to have earned two silver stars and five bronze stars. So either there are somehow secret classified medals, which seems unlikely, or he lied.
We also know one other incident in the book was likely not true, because Kyle lost a defamation lawsuit over it, and the gist of the suit was that the entire incident never happened. This and a couple of other unlikely incidents he reported are the subject of a very interesting Snopes article, the verdict of which was "Mostly False"*.
If he exaggerated his medal count, which is easily traceable, and we know one other account in the book was at least partially made up, and a couple more outside of the book, it seems incredibly naïve to maintain that the rest of his account in that book, which would be less traceable, was entirely truthful.
* - h/t to Steve Bird in the comments