Zubairs, visa sponsorship means that an employer is putting together a package of documents (usually with the help of their lawyers), pays the fees and submits the application with the US government for consideration. Now, here's the problem: the US issues only 85,000 H1B visas every year. This year, I've heard there were over 240,000 applicants. If that happens, if there are more applicants than visas, the government conducts a lottery – they select applications they will CONSIDER (not approve, just consider). So, 2 out of 3 applicants won't even get their documents reviewed by the government. Those lucky ones who got selected in the lottery might or might not get a visa still – if there's something about the application that the government doesn't like, you can get a rejection.
Under the US immigration laws, every applicant for an H1B visa needs to be sponsored by an employer (in other words, your Big4 company must offer you a job and be willing to prepare your application and pay the fees). But, the mere fact that your Big4 will sponsor/apply you for a visa does not guarantee you'll get one. You might still not get selected in the lottery, or the application will be denied on other grounds.
Lastly, timing: the applications to obtain H1B visas for 2016 that would allow you to start working in the US in October, 2016 were due on April 1. Since you've already missed the deadline, the earliest you'll be able to start working in the US is October 2017. Just something you should consider when applying for jobs.
AUD - passed
REG - passed
BEC - passed
FAR - passed