This is almost true. You want to seeI am not familiar with this topic, but I remember seeing this recent paper. It claims that if the map has no singular fiber, it is a fiber bundle (all the fibers isomorphic). Moreover, if the total space is algebraic, it can be a product family after a suitable base change, as you claimed.
Edit I made my answer more informative, as the comments below suggested. I thought the abstract of the paper would answer to the question.
