How to stop newly created leads from entering the nurture programs?

You are on the right track. You need to figure out what the common criteria is between the leads that you want to prevent from entering nurture—could be a certain value in an existing field field like Person Source, or a new field that you need to create and populate using hidden website fields or some other manner. Then you need to set up your nurture so that people with that certain criteria are not added to it.

Without seeing your exact setup, I am assuming you have a Smart Campaign to add people to the Engagement Program, maybe called "01 Add to Stream", and the Smart List trigger is "Person is Created." If, for example, you wanted to prevent people who have Person Source = Sales Sourced to the Engagement Program, you would add a filter "Person Source is not Sales Sourced" to the Smart List.