Some have immersed themselves in internet materials that magnify, exaggerate, and in some cases invent shortcomings of early church leaders. Then they draw incorrect conclusions that can affect testimony. Any who have made these choices can repent and be spiritually renewed.
Elder Quentin L. Cook, General Conference, October 2012
There are exaggerations and misinformation out there but those I have known who researched Church history using LDS sources have found much that troubles them, and once they found that the official history and actual history do not match the flood gates of doubt were opened.
For me this started before the internet, much less google, existed.
So when I started to go back to an LDS church I had to study and be at peace with all the issues that troubled me, like the priesthood ban and the curse of Cain doctrine. So getting to the bottom of these issues is what brought me peace with so many troubling issues. Rock Waterman at PureMormonism helped me get back to church and stay.
And as the link to Rock's post will testify, we need not have a testimony of the Church, only of the Gospel, Jesus, and scripture. The Church is an institution and is neither true nor false but the Gospel is what I believe to be true.
I counted once, out of curiosity, during fast and testimony to see how often I heard "I know this Church is true" versus "I know the Gospel is true" and it was 8 and 0 on that count. No one professed their testimony of Christ they instead professed their faith in leaders from Elder's Quorum to Relief Society to the President of the Church himself. Christ was only brought up when these people finished their talk and prayed in His name, otherwise He just wasn't important enough to bring up.
To me asking whether or not the Church is true is like asking whether or not I think BYU is the only true college.
Our Gospel is distinct in the world, and it is that Gospel that I do have faith in. I don't think our leaders are bad people, I think they're just people as fallible as I am so the only time I put absolute faith in anything is when I put my faith in Christ and in the Gospel.
The Church and the Gospel are two different things as found in Ronald Poleman's 1984 conference talk here and here.
The questions as to the authenticity of the Book of Mormon and the rest of LDS scripture are heavy and hard to hold up, they, like all scripture must be taken on faith. Because we will never be able to prove the authenticity of any scripture, the Bible included, to an empirical level. There just isn't the evidence to prove it, that's why faith matters.
You cannot prove, even with science, why it is you are alive, science can't accurately define consciousness so what we call our true identities or personalities are not proven to be of God or a happy accident of random biological evolution.
We can not prove why we live or what, if anything, happens after we die no matter how many reports of an afterlife we discover, without tangible testable proof we have only our faith.
I think when it comes to doubts about the church, explore them until you reach some level of satisfaction as to the answers you've discovered. I would recommend prayer and asking for guidance and validation from the spirit as to any teaching you come across, if it troubles you dig deeper until you find an answer you can live with.
Many will say you must find the truth, and that's not bad advice it's just that every Muslim, every Catholic, every Jew has a testimony of their Gospel and they are as certain as you've ever been about yours. So from a different perspective the same truth may look very different. And there are competing faiths to look to, so what I think is important is not so much what your beliefs are about God, and whether or not you love Him but whether or not you show love to all His children is what I think matters more.
Doubts lead to questions but questions don't always lead to answers. Faith has to be front and center in any religion. Just because you know the flaws of our Church historically or the flaws of our leaders does not always mean that the Gospel itself is flawed. The church is run by human beings who don't always agree with one another which is why you often find discrepancies in what is taught over the pulpit and you find much that contradicts in scripture but I would argue that much of those issues are rather counter-intuitive rather than simply contradictory.
They say unto him, Why did Moses then command to give a writing of divorcement, and to put her away?
8 He saith unto them, Moses because of the hardness of your hearts suffered you to put away your wives: but from the beginning it was not so. Matthew 19:7-8
Jesus lets us know that man wrote the law of divorce that it was not of God. So what else did the hands of man sully, what else was the product of men and not of God in scripture?
The 613 commands of Mosaic law are very troubling at times but they have an eerie parallel with the Code of Hammurabi which was seen as a solid base of laws to live by in the ancient world and hard as it is to believe, the Mosaic law version was actually very progressive and comparatively lenient.
What helped me hold on to staying in this church has been a lot of prayer and study and friends like Rock Waterman who has an unorthodox but Rock solid testimony of this Gospel and has helped on numerous occasions to talk me through my doubts, to explore them and to reach a resolution I can live with.
For me doubt is like when you loose control of you car in a spin, to regain control you need to steer into the skid. Study more and more until you discover the answers that you can live with, you have to live with the consequences of your actions and decisions, no one else. Therefore, you need to be fair and true to yourself in your life where ever that leads you.
