The cross Bronx is an absolute horror show. Tractor trailers everywhere, cars, trucks, and taxis that absolutely don't give a hoot about you and your trailer wanting to get over. You putting your directional on to change lanes is the signal to other drivers to speed up and not let you in. I don't know when you plan this trip, but I would avoid nyc, unless that's your destination, like the corona virus.
What I would suggest,is, and this is assuming youre taking I78 to I95, would be to stay on I78 past 95, and take that to the Garden State Parkway, (toll road). Take that to the end, which is the NYS thruway (I87/287) and take that east to the Tappan Zee Bridge. When you cross the bridge, you want to work your way to the left lanes, because 87 and 287 split, and you want to be on 287. tTake 287 to the end, and pick up 95 north.
Your truck and trailer will thank you.
Kevin