Try www.kidrex.org It is basically Google but is supposed to be filtered to a greater extent.
I teach in China and I have all ESL students so I made a list of web search resources for my students. They are listed on my website for my students; www.apriln1988.weebly.com Feel free to check it out... or use it. It is a beginner's level website as I am not a web designer, so have patience! I'm working on it ;)
WorldBook Kids is a great source for students to use when researching. My school district subscribes to this site. Check with your school's librarian for other resources that are available in your district.
As now-a-days kids are becoming sufficiently net savvy with a bit of parental help they can grasp the basic search with ease. Your search for search engine specially made for kids clearly shows that you want your kids to browse safe content. Ask Kids, KidsClick, Quintura for kids, Boolify, KidRex, Study Search, Yahoo Kids, are some of the teen ready-made search resources that are completely safe for your young ones.