The burden of proof is on the other guy... it's hard to prove a negative. (but the joke is well-taken).
Please catalogue your evidence with citations whenever possible, and make sure you use reputable sources. Please also disclose your precise statistical procedure and experimental setup so that we may assess the validity of your findings. The questionnaires that you have given to women to assess their satisfaction are also of interest, as changes in wording can elicit different responses. I hope your experiment procedure has controlled for that.
Person A: "I observe a bias".
Person B (me): "Please show me the bias. I haven't seen it."
You (to me): "Prove that you don't observe a bias".
Christ, the level of discourse in these conversations is abysmal. We've reached the point where asking for citations is met with the response of "citation needed".
Sure, you can nitpick over the use of the word prevalent, but it's quite clear from the context that the conclusion was drawn on basis of the experience described (other than the article).
Anyway you want to cut it, this is a completely pedantic derailment of the original point: OP is making completely empty declarations.
You didn't only disbelieve his statement, you offered a different claim.
If things are not equal, you can still prove that there is no bias by asking the disadvantaged group for their opinion. It's not like polling this group is impossible.
Statistics do not work like this. You either positively reject the null hypothesis (i.e.: "these things are different") or you fail to reject the null hypothesis (i.e.: "I cannot assert these are different").
Formally speaking, I am doing the latter.
Informally speaking, everybody else understood my point as a rebuttal to OP's. More explicitly: "claims of X seem grossly exaggerated" should be taken as "I haven't seen compelling evidence for X" -- it may be awkwardly phrased, but I suspect you know exactly what I meant.
You can also just outright swap around your null and alternative hypotheses for a frequentist-style test. For example, your null hypothesis could be "proportion of women to men is 0.1" and your alternative hypothesis is that it isn't, for a standard two-tailed test.
* Well, other than maybe the word 'belief', but this isn't clear imo
In which case I'll reiterate my request that he provide examples, since I've not observed the effect he seems to see all around him.