No, you cannot work in the U.S. on an H-4 visa. Your options would be to try to secure work authorization by changing status to a nonimmigrant visa category (as the primary beneficiary) that permits work.
Consult an attorney. Find a job at a college, university, or affiliated institution. Have the attorney determine whether the employer is cab-exempt under the H-1B visa program. Then file to change your status to H-1B. Use premium processing, not only because you will be able to begin work within 3 weeks or so, but because of enhanced access to the CIS during the pendency of your application. Under a memo issued by CIS in Dec 2006, time spent in H-4 status no longer counts towards the 6 years of status you get on H visas in general, meaning you could get a fresh 6 years from the time that your H-1B is approved.
