Amid charges that she is blatantly tipping the presidential primary scales in favor of Hillary Clinton, Democratic National Committee Chairwoman Debbie Wasserman Schultz is coming under fresh fire from progressives who want her removed from her post immediately.

The progressive advocacy group RootsAction said Tuesday that a 4-day-old petition calling for Ms. Wasserman Schultz’s ouster had received 23,000 signatures. Ms. Wasserman Schultz’s detractors argue that she has wounded the Democratic Party’s credibility by essentially rigging the presidential primary process in favor of Mrs. Clinton.

“The head of one of the two big political parties in the United States is trying to manipulate the presidential election process by limiting direct debate and tilting the national party apparatus in favor of one candidate. This is unacceptable,” RootsAction.org co-founder Norman Solomon said in a statement.

For months, the DNC — and Ms. Wasserman Schultz in particular — has faced charges that it wants to make the primary process as smooth as possible for Mrs. Clinton while limiting opportunities for other candidates such as Sen. Bernard Sanders of Vermont, a liberal favorite.

RootsAction and other critics say specifically that Ms. Wasserman Schultz has used the party’s primary debate schedule to reduce media exposure for Mr. Sanders and the party’s other presidential candidate, former Maryland Gov. Martin O’Malley.

The DNC is holding six official debates this primary season. Several of those have been held over weekends, when TV viewership typically is lower. The next debate is scheduled for Jan. 17, a Sunday.

“The results, thus far, have been bad from the perspective of Wasserman Schultz’s own party, including domination of the media’s extensive election coverage by the other big party. Another result has been reduced exposure for the Democratic candidate polling strongest against Republican rivals, Bernie Sanders,” RootsAction co-founder Jeff Cohen said.

Critics say another example of favoritism toward the Clinton campaign is the fact that the DNC last month temporarily cut off the Sanders camp’s access to crucial voter files kept by the party.

The ordeal began when a Sanders staffer accessed private data belonging to the Clinton campaign. Although Mr. Sanders acknowledged his campaign’s error, he, along with supporters and critics of the DNC, argued that it was a massive overreaction to suspend his access to the party’s voter file.

Mr. Sanders’ access to the voter file was restored after he filed a lawsuit.