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"making headway on" is correct and usable in written English.
You can use it when discussing progress on a difficult task or project. For example: "We're finally making headway on the renovation project."
Idiom
Make headway.
Progress in what you are trying to achieve.
Exact(43)
"But I think they're actually making headway on that one.
Unlike on those tasks, Kushner appears to be making headway on yet another one: creating jobs overhauling the tech of the federal government.
Meanwhile, others are making headway on fund-raising.
He added that the opposition "is making headway on the ground".
That's why the Democrats are having a tough time making headway on the economic issues.
As trends go, chicken potpie has been making headway on menus since the beginning of the recession in late 2007.
Similar(17)
The N.F.L. has made headway on college campuses, too.
"The other big question we hope to make headway on is what is the dark universe….
In September, Marclay was back in London, struggling to make headway on the manga book.
We'll never make headway on our serious challenges without robust national discourse.
Meanwhile, I've been trying to make headway on the adultery front, with some hopeful indications.
More suggestions(3)
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com