Somehow as soon as possible in the application/recruitment process, let the potential employer know that although "sponsorship" is required, the process can be quick, cheap, and easy.
SUPER easy: https://www.uscis.gov/working-united-states/temporary-worker...
You do not need a lawyer and the letter of employment can be done by the HR department of your employer, just be sure to include everything they ask for. I did it all by myself and got my company to reimburse me the fee after I came in to work.