I noticed that many places, specially in the United States, have fences separating two streets, even for pedestrians. My guess that the streets are not connected for cars to calm the traffic on residential streets and to relieve congestion on main roads, but the segregation for pedestrians also seems more to intend to lengthen the trips for people walking, thus forcing them to use cars instead. Since both sides are public spaces, thus crossing those forbidden passages does not seems to constitute trespassing at first, but would that constitute another violation?
"Public space" is not a relevant criteria when considering trespass or other crimes/torts against property. The relevant criteria is who owns it and what they allow you to do on it.
All land in the USA is owned by someone. That someone may be a government; that does not make it a public space - Camp David is owned by the US government; it is certainly not public.
The owner of the land can decide (subject to the law) who has access to their land and in what circumstances. If they erect a fence then they are saying "You cannot access my land here" - if you ignore this then you are trespassing. This is true even if there are legitimate ways to access the land i.e. there is a place where there isn't a fence; to avoid trespass you would have to access the land from there. If you think of this in terms of a public building like a courthouse you are free to enter through the unlocked front doors but not by climbing through a window.
The trespass is in the act of crossing the fence - that is the act that you have been implicitly denied permission to do. Being on one side or the other is not trespass.
For the specific image that you show it is quite likely that those roads are owned by different people - the highway is probably owned by the state while the cul-de-sac is a local government road.