CAIRO/JERUSALEM — Israeli troops and tanks pushed on Saturday into parts of a congested northern Gaza Strip district that they had previously skirted in the more than seven-month-old war, killing and wounding dozens of Palestinians, medics and residents said.

Israel’s forces also took over some ground in Rafah, a southern city next to the Egyptian border that is packed with displaced people and where the launch this month of a long-threatened incursion to crush hold-outs of Palestinian…