The thing is, almost all postdocs start on J1 visas, not H1b. Most schools hold out the possibility of moving you over to an H1b as an incentive after a few years. It's also complicated by many J1s come with the rider that they must return to their home country for two years and cannot shift to an H1b. There is a pathway to apply for a waiver from the State Department, but I'm told these days the process takes about 18 months to get the waiver - BEFORE you can be eligible to apply for an H1b.