Ok so here is how I deal with such stuff:
1st, I know Islam is correct. I looked into other beliefs(Judaism, Christianity, Buddism, agnostic, etc...) and I realized this religion came from god. This religion made the most sense to me.
2nd, If you believe that Islam is correct, then evertime you find something that doesn't feel correct to you. you don't say that there is a problem with Islam. You know one of the names of allah is the Just. And since Allah brought this religion then you blame/accuse yourself, other people, or the culture that made you think this wrong.
One simple example: Now its becoming legal that gay marriage is allowed. If in the next decade seeing gay couples walking together is normal, does that mean Islam is false because it didn't allow gay marriages? You see the point ? Same thing with 4 wives or marrying at a young age(Not to mention it has been practiced by non-Muslims too), or having slaves(Common thing between non-Muslims, and Islam cane to stop it). Just because people has stopped practicing these things nowadays doesn't mean Islam is wrong. But it is the culture or the people who are wrong. Culture keep changing towards the worst. People are moving away from the human nature we were born with.
Other than that. In Islam it is encouraged to ask question. I don't like it when some people try to doge these questions by saying this wasn't authentic, or the prophet was not like that etc... Allah in the Quran say:
And We sent not before you except men to whom We revealed [Our message]. So ask the people of the message if you do not know. http://quran.com/16/43
Interpretation, Ask those scholars or knowledgebale people if you don't know. Coming to this website and asking this question is a good thing. Al-Hamdolelah. This is why we have this website and other websites on the internet. May allah reward you. If you see anything else you want to learn more about. Please feel free to ask here. Here is an example of a controversial question that has been asked which might help answer some of the claims on that website Is it true that Islam is a religion of peace?
Last advice, It is good to know about these controversial stuff. But I highly recommend you to stop visiting similar sites that only show controversial stuff and ignore the amazing teachings of Islam. Instead, dedicate your time to be a person that Islam wants, a person who have good manners i.e: Kind, merciful, Treat your parents and family well, etc...
UPTADE: To answer the claim regarding Um Qirfa. The story mentioned in the anti-Islam site is true but not all parts of it. The part that talks about putting her legs to rope is not true. Scholars said this is a really weak narration. Another point, Not mentioning the more authentic saying of the prophet regarding the mutilation of the dead bodies, even though they claim to know a lot about islam, killed their credibility. This is where I stopped reading the article.
Here is the hadith:
.....do not embezzle the spoils; do not break your pledge; and do not mutilate (the dead) bodies; do not kill the children
the 3rd point, The prophet didn't ask to kill her specifically. He asked "Zaid bin Harith" to attack the people of Um Qirfa because she asked more than 40 of men and their children to assassinate the prophet. Hence, the prophet told the muslims to stop them. When Ziad reached there with the muslims, the two groups got into a battle and Um Qirfa was killed there because she was in the battle with them too.
Source: [ARABIC] Islamweb - Um Qirfa incident