Happy family - can I sponsor them?
“I AM a naturalized American citizen. I have been a citizen for nearly three years. I married an American woman 12 years ago and we have since divorced, after having one child. I have been seeing an Irish woman for the past year. She lives in Ireland with her young daughter and son – she too is divorced – and would like to relocate to the U.S. My questions – what would their legal status be if they wanted to come out and live here for a while? And if we decided to marry – we are headed in that direction – how would they become legal?”

Your girlfriend and her children could come to the U.S. for up to 90 days under the visa waiver program. If they wanted to stay longer they could apply for B tourist visas at the American Embassy in Dublin, which would potentially allow them to stay for six months with the possibility of an extension.
Presumably, if she has young kids she would want to travel here during the summer months when they are off school, so the 90 day visa waiver should suit the family fine.

Those who obtain U.S. citizenship through marriage to a U.S. citizen and then divorce, such as yourself, must wait five years after the issuance of the green card in order to confer immigration benefits to a new foreign spouse.

That deadline has clearly passed in your case, so if you marry, you would be able to sponsor your girlfriend and her children for permanent resident status. However, the father of the children may well have something to say about that which is another issue you’ll have to sort out prior to applying.

Presumably there is some sort of custody arrangement in place between your girlfriend and the children’s father.

After you marry, you would be able to sponsor your new wife as an immediate relative. The waiting time for processing is a few months to over a year depending where the application is filed.
Once you are married, U.S. immigration law would consider you to be the stepfather of her children.

As such, you would also be able to sponsor the kids as immediate relatives.

Stepchildren can receive immigration benefits as long as foreign parent married the U.S. spouse prior to the stepchild’s 18th birthday, which applies in your situation.

Read more:

Can US citizens help get their Irish children visas?

Green Card - Is my old Social Security number still valid?