You have a couple of options.
Depending on what you're trying to do, you might be able to model this using an added cost point barrier in your analysis. Here is some documentation about barriers: https://pro.arcgis.com/en/pro-app/latest/help/analysis/networks/barriers.htm
If that isn't sufficient, and if you have a Streetmap Premium license (I'm assuming that's the Esri proprietary data you're referring to), you can talk to your Esri representative about getting the Custom Roads version of Streetmap Premium, which is editable. You could then edit in one or more turn features at the intersection you're modeling to add additional travel time when turning left, going straight, etc.
