Well, they have been brainwashed.
I completely understand why they still believe because I used to believe too.
These are a couple of the things that I have always told myself to excuse my TBM family members and my friends in the past, but I'm getting really tired of excusing them, and am becoming a little bitter towards them for their cowardice.
I know this isn't very healthy but maybe these feelings are just part of the recovery process.
I understand that the church is a gigantic and powerful propoganda machine, but at what point am I allowed to stop giving them a pass because they have been brainwashed, and start disliking them for being such head sand burying cowards.
It used to be that back in the day nobody knew about these lies. Now the information is out in the open for everyone to see and they won't look.
There is no longer much of an excuse for them. How can you know that the Book of Abraham is a fraud, that Joseph Smith married teenage girls, that Joseph married other mens wives etc... and still go to church, teach it to your kids, support missionary work, bear your testimony etc?
Even one of these facts on its own should be enough to make you honestly reassess your beliefs. But when this is just the tip of the iceberg?
When I was a member I didn't know these things, and when I found out I knew that I couldn't remain intellectually honest and stay, and I was only seventeen!! My TBM family and friends now know about these things, they have read Rough Stone Rolling, they have been exposed to most of the other facts, yet they still ignore everything that they have learned and carry on like nothing has happened! It is maddening!!!! It makes you hate humanity for being so spineless.

How do they live with themselves? I think the social network is nice, and I often miss being a part of it, but the benifits of being a part of that network are not worth being a liar.
I would bet that at least a quarter of the members know about this stuff by now, and they still do nothing! They ignore it and pretend like it doesn't exist.
How can they ignore it?

I understand your frustration. It's exactly like that story in the bible where people were sick and dying and Moses (I think) puts a stick up in the air and tells the people if they just look at the stick, they will get well. But because of the "ease of the way" the people refuse to look and continue to die.

Today, the stick is the internet and if you hear something about the church that makes you say "WTH?" you can check and double check the facts. I really don't understand people who don't want to know for sure who and what they are following, who and what they are dedicating their lives to. When I read that there were no religious revivals recorded around Palmyra in 1820, my first reaction was "They are LYING to us." Then I became obsessed with finding out the truth. There was too much at stake and the church demanded too much of me to shrug off such an obvious problem. I mean, historically, there is no way the first vision could have happened the way we'd always been taught. How could I NOT find out more?

The only thing I've discovered in answer to your question "How could they ignore it?" is that the average member who refuses to look is blinded either by fear or ego. Either their pride and sense of accomplishment are so wrapped up in Mormonism that they can't separate it. Or they define themselves as special, Mormon, chosen so deeply that they can't bear to think of themselves as ordinary or worse, fools. Fear is even more crippling. Bishop Jackwagon grew up in a very disrupted family which made him terrified of losing his family when he married and had children of his own. Once, I was joking with him about his youngest and how in a couple of years, he and Sister Jackwagon would be empty nesters and you should have seen his reaction. He looked terrified - like I pulled a knife on him. Since then, I've noticed how many of his comments and messages in the ward bulletin reflect his clinging to his family and what he's done to keep his family together forever. The idea that he doesn't have a magic key to an eternal family would probably make his head explode. In fact, that's probably why he exploded at me when I started parading out my facts against the church - I was scaring him to death. If you are so afraid of losing your family, or failing, or what the neighbors think, or going to hell or whatever, it's awful to face those fears. Once you face the fears though, you can face the truth about the church.

Sometimes for people who live in heavily infested areas the resulting fallout is too much., They risk losing jobs and standing in the community. They risk having friends and neighbors becoming enemies and strangers, and then there is the threat of divorce and family estrangement.

In the Beehive, if you were to be seen as a hornet, you would be destroyed immediately. The church ingrains the beehive mentality from cradle to grave, though it usually promotes the ideal ao working to a common goal, the threat of the enemy is always in the front of the collective consciousness.

And then there is the believer who recognizes the workings of the spirit in their lives and identifies themselves as loved by God as administered by the church, yet are unable to fathom that this is entirely seperate. The church is a familiar vehicle for their faith, but it does not control the spirit nor does God look up your profile in the church membership records.

The reasons are as varied as the number of lies the church tells. But from one who knows what it is like to be shunned by family, I would suggest that attacking them or seperating yourself from them because of something as silly as religion is a terrible mistake.