'win them support' is a correct and usable phrase in written English.
You can use it when you want to describe someone or something taking action or making an effort to gain support or approval from a specific group of people. For example, "The politician sought to win them support by visiting their constituency and speaking to local residents."
But continuing to wreak economic havoc is hardly going to win them support.
It's hard to imagine how having a new third party would suddenly cause right-wing Republicans to abandon obstructionist tactics, when those very tactics continue to win them support from their conservative base.
Offshore wind proponents hope the jobs that can be created with the new industry could win them support from Trump, who counts job creation as an important part of his agenda.
The challenge for Mongolia's authorities will be to address the concerns that fuel their anger and resentment and win them support, while limiting their opportunity to do real harm.
This has clearly won them support.
As hard as it would be for the government to admit it, the Naxalites do have some valid grievances that have won them support.
The trio recorded a four-song demo tape, which won them supporting roles at the concerts of artists such as Johnny Thunders and Warren Zevon.
"If they were really seeking change they would resort to actions that would win them the support of the people," the professor said.
She is angry that it took a media storm to win them more support, and is anxious to make the point that her family's difficulties are shared by thousands of families with disabled children around the country.
But the earliest efforts by GOP lawmakers to tackle immigration policy on Capitol Hill aren't likely to win them new support among Hispanics.
But there is a risk that the airstrikes will embolden the rebels and win them broader support in Yemen, where many are opposed to any kind of foreign intervention.
