working as an SEO, is it necessary to know how to make the website browser compatible? Or it is the work of developer? One thing more I would like to know is that how to make long URL of Dynamic websites Short? Thanks in advance
As an SEO (are you?) you usually don't need to worry about browser compatibility: it's the designer's job. There are various ways of shortening your URLs, depending on the case and on what type of server you are on (URL rewriting, 301 redirects, etc.).
Make the website compatible with most common web browser is web designer's job, your duty just think about the marketing and SEO for your website .
Try Browsershots.org and Seo-browser.com They let you see how visitor and Search Engine view you site.
Browser compatibility is the job of the designer, not the SEO. In any case if you are interested, you can offer that as part of your seo service. Then you can use sites like crossbrowsertesting.com to test the website in different browsers as to how it appears.