Suggest any Free Website that helps with checking Google.com and bing.com SERP with accurate results?
try this: https://tools.digitalpoint.com/tracker it can be found under the TOOLS option at the top of this page.
Both of the above tools are of not much use. The first one offers 1 week trial and DP tool offer top 20 SERP. Require more suggestions
There aren't any free tools that will offer good results over a long period of time. For paid I suggest a service called serpfox. They aren't that expensive and do a good job.
I have tried semalt.com and its useful to check my rankings but it only give an temporary account for 6 days from my start of registration.
If you need a desktop solution - I suggest you to try free Rank Tracker by link-assistant.com http://www.link-assistant.com/rank-tracker/ If you need a Web-based tool for rankings check, WebMeUp is what I consider the best option: http://webmeup.com/
You may like this tool ( http://serp-checker.ezmlm.org ) to check your SERP ranking for Google, Bing and Yahoo. This site also have feature to check SERP for a specific google domain.
You can go 200 deep if you are a premium member (although not sure there's much value in anything ranking beyond 20 anyway).
Don't forget that with any FREE SEO tools the results won't be as effective as you wanted... That's why people did their HARD job and created such scripts or tools and they would charge a small fee to allow you use their tools... With free tools, you could only see the DEMO version if I can say so!
serpfox.com proranktracker.com They both allow to track up to 50 keywords for free, both are awesome try them out.