Opposition councillors from the Lib Dems put forward a motion asking members to support them in calling for the resignation of the Shropshire Council cabinet member for highways, Steve Davenport.
At a meeting at Shirehall on Thursday, Lib Dem leader Roger Evans said: "This is not something we normally do but the general feeling is that you should resign and let a new person look at the problems facing this county.
"The vast majority of councillors here have raised highways problems that are still unresolved.
"We have made the national news because of the state of our roads. The council's reputation has been trashed.
"Steve, I am sorry, but please resign and maybe take over another portfolio. It needs fresh eyes to look at it."
The motion was backed by fellow Lib Dem councillor David Vasmer, who said the highways department was in "absolute chaos", with good staff resigning because they are not appreciated.
However, council leader Peter Nutting rejected the calls.
He said: "I think this is very childish. I think it will undermine the work that's going on in highways.
"This is just pub politics because there is an election next year - let's all just grow up.
"Steve has my total support."
Members voted to throw out the motion.