Most bassinets are designed for babies up to around 4-5 months old or until they start rolling over, whichever comes first. At 4 months your son is pretty much at that boundary, so it's not crazy that people are nudging you toward a crib. That said, every baby develops differently - some roll earlier, some later. The real safety issue is when they can flip themselves over consistently, because then they might end up face-down in the bassinet's soft mattress.
The transition doesn't have to happen overnight. If he's not rolling yet and you're comfortable with the bassinet setup, you've got a little wiggle room. But most families find it easier to move kids to a crib somewhere between 4-6 months just to avoid the rush if development happens faster than expected. A standard crib is safer long-term anyway since it's built for older babies and toddlers.
One thing to check - make sure your bassinet's weight and size limits haven't already been reached, and that the mattress is firm (not soft or cushy, which increases suffocation risk). If you're going to stay in the bassinet a bit longer, that matters more than the age number. When you do move him, just pick a time that feels right and give him a few nights to adjust. The crib will be fine whenever you make the switch.