The proposed mass extinction due to Great Oxidation Event is actually an elucidation by scientists.
After reading many articles online, although there are evidences of GOE, every article states the extinction of most anaerobes as something that should have happened as a fact (Because $O_2$ is considered poisonous for obligate anaerobes). To better understand this, imagine if there was a spike in Carbon Monoxide in the atmosphere, for Oxygen dependent organisms like us, inhaling $CO$ is harmful and therefore a spike in $CO$ would be toxic and lead to death.
Here are some excerpts, which state the extinction of anaerobes being factual more than anything else:
From Arizona State University, Origins Project >>
In a relatively short amount of time, Earth went from having very
little oxygen to what may be the highest levels of atmospheric oxygen
it has ever had. This event had wiped out most of life on the planet
to which the oxygen was poisonous. Some of these anaerobic organisms
were though to have survived by burrowing into the earth where oxygen
levels were survivable.
From Slate.com - Bad Astronomy Blog >>
At some point, though, as cyanobacteria flourished, the minerals and
other sinks became saturated. They could no longer absorb the oxygen
being produced. It built up in the water, in the air. To the other
bacteria living in the ocean—anaerobic bacteria, remember—oxygen was
toxic. The cyanobacteria were literally respiring poison.
A die-off began, a mass extinction killing countless species of
bacteria. It was the Great Oxygenation Event.
From Imperial College London >>
The permanent introduction of oxygen into the atmosphere fundamentally
changed the ecology of the planet. The first impact was on anaerobic
organisms, which didn’t need oxygen to survive. The rise in oxygen
levels was poisonous to them and led to most them becoming extinct.
Some managed to survive, but they retreated deep into the subsurface
where there is no oxygen. This was one of the first mass extinctions
in the history of our planet, and it was also the catalyst for the
evolution of oxygen-based life – us!
As a result, based on what I have read so far, we really dont have any stone cold proof that rise in oxygen lead to mass extinction of anaerobes. The mass extinction is what scientists consider should have happened as we know $O_2$ is poisonous for strict anaerobes.