For a moderate conservative like me, it is really hard in this world to escape the overwhelming influence of the liberal dominated mainstream media (msm) with ABC news, NBC, CBS, CNN, MSNBC, NY Times, and an overwhelming majority of newspapers in this country. There really is no escaping it's influence. The night before the 2016 Presidential election I was somewhat depressed because all I heard from the msm was that there was no way Trump would win, the poll numbers for Hillary were good, Trump was a terrible human being not fit for office, and was a misogynist pig who treated women poorly. I actually believed that this msm onslaught was so great, their influence so far and wide, and so overwhelming... that was my overall perception of how the world saw a potential Trump presidency. In a sense, I was somewhat conditioned to believe that no one could persevere against the avalanche of vitriol and negativity that was leveled at Trump. The media's message was this was Hillary's time. I was legitimately shocked and stunned (my Mom was pissed) when the election returns came back election night, I thought he had no chance of winning. What happened? It wasn't just deplorables, gun nuts, and a few Charleston morons that voted for Trump. Did you know that in the category of educated women, more voted for Trump than Hillary? My sense after the election is that I had to recalibrate my perceptions of the msm's influence. My sense now is that there is a silent majority out there that does not tweet, does not engage in debates on social media, and most of them do not enjoy talking politics with their friends. But they do pay attention and they do not mindlessly drink the Kool-Aid that the msm routinely offers it's viewers or readers, they take in what is being offered but see it with a skeptical, critical thinking perspective. I believe this silent majority includes people on the left ideological spectrum and on the right. They may not wear their politics on their sleeve, but they vote, and a lot of those people voted for Trump in my opinion. I believe the era of power and influence for the msm is waning and that people see that they all have an agenda and see through all of the lies and misrepresentations. The msm said that a Hillary presidency was inevitable, the American public had other ideas.
First, a general observation - it might be hard to believe, but a self-described "moderate conservative" in America is probably closer to a far right conservative in most other places in the world. That's important because it informs your view of what "liberal dominated" means. For example, you mention the New York Times as an example of liberal dominated mainstream media. The Times is one of the most respected English language newspapers in the world. they've been winning Pulitzer prizes for decades and have broken massive, consequential stories again and again. They employ a number of right-leaning columnists, like David Brooks, Ross Douthat and Bret Stephens. They routinely publish guest columns by other notable conservatives. Most importantly, they strive to adhere to journalistic ethics. They're not perfect. But they try hard to be fair. Perhaps you are not quite as "moderate" a conservative as you think if you view the Times as wildly left wing.
As for the election, polling indicated that Clinton was leading. Less than two weeks before the vote, Comey went public with the revelation that the FBI had reopened the Clinton email case. He did that while remaining quiet about the fact that the FBI was investigating the Trump campaign for Russian meddling. The Clinton revelation occurred too late for opinion polling to catch up.
Regarding Donald Trump, there is a serious disconnect between camps. When, for example, the NYT criticizes Trump for saying or doing something stupid, or mean, or untruthful, it seems like proof of bias to those who support Trump. But almost all of the time it isn't bias, it's factual reporting. It just appears like bias because Donald Trump is almost continuously doing or saying something stupid, or mean, or untruthful. Trump is uniquely awful, and to report honestly on him looks like anti-Trump bias. It really isn't, but if you lean far enough right to perceive the NYT as strongly left, you may see bias where there is none.
Finally, educated women and Trump. I do not for a second believe that you intended to deceive here, but that statistic is wrong. Among college educated WHITE women, it was 51% Clinton, 45% Trump, which is fairly close. However, Hispanic, non-White and Black college educated women voted overwhelmingly for Clinton. Now, Trump has bragged that he won 52% of women's votes, but that is incorrect. He won 52% of WHITE women's votes. Overall, American women voted 54% Clinton, 39% Trump.