What major political events happened in 1998?
The most important political event of 1998 was the ongoing Bill Clinton-Monica Lewinsky scandal. By the end of 1998, President Clinton had been impeached by the House of Representatives. The trial started early in 1999 and Clinton was, of course, acquitted.
In a development that may have been related, the Republican Party lost ground in the House of Representatives in the 1998 midterm elections. That led to the ousting of Newt Gingrich as Speaker of the House.
A political event that seemed important at the time but has not turned out to be very important was the election of Jesse Ventura as governor of Minnesota. He was a former professional wrestler and total political newcomer. But his election did not really lead to any sort of major change in politics.
I don't know if this counts as "politics" for your assignment, but this was also the year in which two US embassies in Africa were bombed by Al Qaeda. The US was also in an ongoing dispute with Iraq over their weapons of mass destruction program. Both of these events have come to have much more importance now that 9/11 and the Iraq War have happened.