Its not easy as it sounds. Being a single that time, I have transferred my H1B myself and it was a nightmare. I had sleepless nights and uncertainty. My GreenCard process was reset. Any H1Bs who knows about the system really well will not risk transferring because of the risks to their green card process.
There is no honest company which pays a fair salary for H1Bs. If you see the data from here:
https://www.foreignlaborcert.doleta.gov/performancedata.cfm
I have a huge circle of friends in many companies. We discuss our salaries and issues openly. Not one of them says that they are paid equal to their American colleagues. Moroever the promotions and salary raises goes to American colleagues, while the H1Bs bears it all to support the family with the income where his/her spouse is not allowed to work (H4 visa)