I know exactly what you mean. I think there's this paradox... you can't fully enjoy life unless you realize that bad things happen, and someday you will die. But if you focus on only those things, you won't enjoy life that way either. I believe there's something built in to our brains that stops us from fully feeling the horrors that happen to others, because it would drive us crazy. I've definitely noticed it getting to me more when I'm depressed though.

You should watch the music video for Brett Dennen's . The words and the video together illustrate what you're talking about in such a powerful way, I find myself watching it over and over. This video may be a trigger for some. For myself, sometimes it makes me incredibly sad, other times uplifted. It depends, I guess, on whether you focus on your own guilt at being like the people in the video, or your insight in realizing it, unlike the people in the video. Knowing what you know is half the battle. You can't help everyone, you can't make the world perfect. But you can pick your battles. You can donate money to no-kill shelters if that's what bothers you most. Or volunteer at a soup kitchen. Or go downtown and just talk to homeless people. Pick a few, and know that you're making someone's life better, maybe even saving a life.
Long ago I realized that though you may not be able to change the world, all you need to do is change someone's world. We all see things a certain way, and if you can help someone change the way they see the world in a way that will help them, then really, you have changed the world. That's why I always wanted to be a writer. I wanted to write one of those books that makes people realize they're not alone, that touches people on such a personal level that it changes their life. Because that's the most that most of us can do.

Another thing that helps to deal with this I think is what Dennen's saying here... Things have always been this way. It may be hard to see, but if you study history, you'll see it. People have always been oppressed. People have always had mental health issues. The world is constantly changing, but the big things always stay the same. I read a lot of books about Tudor era England. Back then, people were being burned at the stake for worshiping the earth or practicing herbalism. Still, if the king wanted to know his fortune, he'd go to a heretic. People lived in poverty with no chance of escaping, because they owned nothing themselves. The king would choose his favorite courtiers to be well-paid advisors. Advisors would betray the king if it benefited them. Pregnancies were hard and many children didn't live to adulthood. The Church told people how to live, and didn't even allow them to read the Bible.
You can look at these details and say the world today is totally different. But how different is it really? We still mistrust the people in charge, accuse them of playing favorites. People still betray their country for personal gain. Instead of the plague and dropsy, we have ebola and cancer. Instead of burning people for their faith, we bomb them. We have social welfare that often just keeps people in poverty. We still have black markets that supply things people couldn't get anywhere else. Babies still die, mothers still cry when they send their sons to war. Nothing ever really changes.
To love and be loved is the best we can do.