The position of strong atheists — myself included — is that we know that no God exists as confidently as we know any moderately complicated but well-established scientific theory, such as evolution or quantum mechanics.
If one objects to strong atheism it is either just because we are confident, or because we assert a degree of confidence that is not warranted by the actual facts. To object to confidence per se entails epistemic nihilism. If one objects that the confidence is not warranted by the facts, then only the substance of the argument is relevant.
The whole argument is complicated (again, much like a moderately complicated but well-established scientific theory) but it can be briefly summarized.
There are two classes of definitions of god: empirically falsifiable and metaphysical. I adopt Popper's general stance that there's nothing inherently disreputable about metaphysics per se (although there can be bad metaphysics just like there can be bad scientific theories).
The falsifiable definitions of God are actually false: they either make predictions that are definitely falsified by experience (e.g. the Problem of Evil) or they are over-elaborated and overcomplicated restatements of ordinary natural science and fall victim to Occam's razor.
The metaphysical definitions of God either require presuppositions at least as controversial (if not more controversial) than simply assuming the existence of God (e.g. the Ontological argument) or are (again) over-elaborated and overcomplicated restatements of metaphysical naturalism and again fall victim to Occam's razor.
The analysis is somewhat complicated by the sheer number of patently invalid arguments for the existence of God. Invalid arguments demonstrate nothing per se (nb. the fallacy fallacy). However, valid arguments ought to drive out invalid arguments; when one position is riddled with invalid arguments, that's strong evidence that there are no valid arguments to drive out the invalid.