Bing is choosy when it comes to indexing and ranking. Maybe you site's content is enough in the their eyes.

Have you submitted your site in Bing yet?

Submit your website to bing webmaster tool. Using Crawling, Caching and Indexing.

Use bing webmaster tools. Submit a sitemap containing all your web pages. Try to build a strong backlink campaign off and on your site and try to rank high for your particular keyword or your site. And over time you should see better results. Als,o make sure your site is structured well for when search engines crawl it.

How old is this site? Everything involving SERPs is a bit of a waiting game. Even if you do submit it to Bing, it doesn't mean that they are instantly going to update the SERP pages. There are currently more than a billion websites and yours is probably not their #1 priority.

When you say you type in your website name, do you mean you are typing it in including the .com (or whatever the TLD is)? If not, there may be plenty of other sites that are better than yours for your domain name without the.com.