Richard Cordray, director of the Consumer Financial Protection Bureau, faced renewed questions Thursday about the legitimacy of his agency.

Cordray, in a semiannual report to Congress, touted the bureau’s accomplishments in drafting rules to fix mortgage servicing and supervising the previously unregulated non-bank financial firms.

He added that the bureau is working on rules to bring greater transparency to prepaid debit cards and is fine-tuning its consumer complaint database — as of Sept. 3, the agency had received 72,297 complaints about mortgages, student loans and credit cards.

But the remarks did little to satisfy Senate Republicans, who worried that the agency’s authority could lead to a reinterpretation of established federal laws.

Several GOP-sponsored bills aimed at stripping the federal consumer watchdog of its authority are sitting in House and Senate committees, though none has much chance of becoming law this year. Republicans want the agency to go through Congress for funding, rather than the Federal Reserve, and to be run by a five-member commission rather than a single director.

The political wrangling delayed Cordray’s confirmation for months, leaving the agency without enforcement power.

Sen. Richard C. Shelby (R-Ala.) said he was troubled that the bureau could potentially create exceptions to a Wall Street reform law that would ban lenders from charging consumers upfront payment of fees when they take out a mortgage.

“The bureau has interpreted its exemptive authority so broadly that it believes it can just ignore the law,” said Shelby, the ranking Republican on the Senate Banking Committee.

“Congress intends us to write rules around disclosures and to clarify and interpret the laws,” Cordray responded.

Other GOP senators asked Cordray how the bureau will ensure that small lenders are not unduly burdened by costs. The director said the agency has established an advisory authority to review the matter.

Not everyone put the screws on Cordray. Members on both sides of the aisle stressed the importance of having an agency dedicated to weeding out abusive practices in the financial services industry. Many praised the bureau’s “Know Before You Owe” campaign to educate consumers about student loans, credit cards and mortgages.

“The [CFPB] report suggests that we’re well on our way to building an agency that can fulfill its mission, a mission that before its creation was too often ignored,” said Sen. Jeff Merkley (D-Ore.).