Is it normal for a new lower partial denture bar to put pressure and cause pain in my lower front teeth?

Not really: If the denture is new then the clasping may need some adjusting to fit properly, it sounds like it may be tight. The connector bar of the partial should not put pressure on the lower anterior teeth to the point of it being painful. Worst case scenario is there may have been a problem with the impression and therefore the framework doesn't fit properly. You may want to have dentist recheck it.

Maybe: A new partial may bit so well that some pressure might follow. However, this should be able to be adjusted, see your dentist.

Maybe, some only: Depends on the type of design. Often we use the lower front teeth to help support the partial. This may be what you are feeling. But, if it is alot of pressure and you are having pain, then this is not acceptable.You should return to your dentist to have it adjusted until it feels comfortable. There is a learning curve, and you will feel pressure. The question is how much and how long to be normal.

Yes, but....: I would see your dentist for a possible adjustment. I am not comfortable that you have pain.