Never Give a Number.
This is both the most sensible and obvious advice (more specifically it's "never be the first to give a number", but nuance), and the hardest advice to follow. It's blindingly obvious because at best, you guess correctly, and they offer you the salary they would have offered you anyway. It's hard to follow, because it's thrown into every hiring team's "questions nobody feels comfortable calling you out on."
You've already given them a target salary and now they're using it as a guideline for what to offer you. Trying to change it now is entirely ethical, but you're in a much weaker position unless you have a competing offer (after all, they know at some point you were okay with 85k, they can probably persuade you to go back there).
You now have to consider your "Best Alternative To a Negotiated Agreement"
Normally, this comes down to a different job. If you have other equivalent or better offers, or you're confident they'll turn up with a bit more effort, then you can be confident about refusing the offer. If you need to make rent at the end of the month or be homeless, and this is the first job offer you've had since 2010, then I personally would consider accepting whatever deal allowed me to make rent.
The true BATNA is that you take the offer and then immediately send your CV out and round again. If the entry-level average really is 97k, and 6 months from now it's gone up, and you're still stuck at 85k (because initial salary strongly influences long-term salary), the likelihood is that you'll end up quitting to the company that's offering the average, simply because 12k a year is a life-changing amount of money. This sucks for the employer, and it's a pain in the neck for you. So my impulse, assuming that I thought I could get the average salary, would be to avoid that process by declining that specific offer.
How you decline the offer is important. After all, it's arguable that you've basically told them the kind of salary you'd like, they've proceeded to invest time into the hiring process based on that, and you're now trying to leverage that sunk cost into a better deal.
If that's what you'd actually done, that's unethical, and can lead to bad feeling and associated repercussions (theoretically you could be blacklisted by the company, etc.) Of course, what you've actually done is made a common mistake in negotiating (Giving a number), and are now looking to fix it.
Something along the lines of
"Obviously, I'm really excited about working for {companyname}, and that offer is definitely looks great in principle, my only concern is that there are other opportunities I'm considering that have a higher entry-level salary, and as we plan to be working together long-term, I don't want to be pressured to move on for financial reasons. Can I just confirm whether we could agree to a review at the end of my probation period, once I've proven to be a valuable member of the team, to a salary in line with {number}?"
might be the way to go (although someone more skilled than me could improve it by probably $100k). But there is the obvious risk that they'll go "nope, don't want to hire someone who'd consider other jobs, thanks for your time".
I wouldn't mention any particular site, because that's just giving them a chance to discredit the source rather than the merits, simply treat the hypothetical average as "a potential opportunity" (which it is) that if asked, you "can't discuss"(because it would be a bad idea for you to do so).
And note that {number} can be higher than your original 95k, because you're not offering the same thing; specifically, you're reducing their risk, because if you turn out to be crap (don't turn out to be crap), they only paid you at the lower level. And if you turn out to be good, a jump in salary is easier to swallow.
Note that the critical thing here is to treat them like intelligent adults who've offered you a very good salary, but one that isn't quite high enough to guarantee your long-term loyalty. And you want to be loyal, so you'd like them to get there.
And obviously, the worst case is that you decline this offer, don't get the job, don't get any other job offers, and struggle for money. So if that's likely to be a major issue (given where you are, who you are, what's going on in your life, what the local job market's like, whether you have friends with a couch), the traditional advice of "don't play chicken with a runaway train" applies.