Considering there is some amount of either wrong, or maybe just misleading, information regarding your broad question which, in turn, can lead to ambiguities and poor and/or dangerous actions, I, a real electrician, will add my hat in the ring backed by the NEC.
Because many people may search and find this post, as it is very generally asked, and in a given situation may opt to run a new grounded circuit to fix this problem and find themselves staring at the exact same situation in their panel, I think it's important to note it is not always wrong to bond the grounded (neutral) and grounding (ground) systems together.
From IAEI, thoroughly explained in NEC Article 250 - Grounding and Bonding. (empasis mine):
The main bonding jumper is one of the most critical elements in the
safety grounding system. This conductor is the link between the
grounded service conductor, the equipment grounding conductor and in
some cases, the grounding electrode conductor. The primary purpose of
the main bonding jumper is to carry the ground-fault current from the
service enclosure as well as from the equipment grounding system that
is returning to the source. In addition, where the grounding electrode
conductor is connected directly to the grounded service conductor bus,
the main bonding jumper ensures that the equipment grounding bus is at
the same potential as the earth.
NEC 250.35 Grounding Service-Supplied Alternating-Current
(A) System Grounding Connections. A premise
wiring system supplied by a grounded ac service shall have a grounding
electrode conductor connected to the grounded service conductor, at
each service, in accordance with 250.24(A)(1) through (A)(5).
Again this is to answer the naive statement that they should never be connected. The National Electrical Code absolutely disagrees with that and removing this connection is indeed detrimental to the safety of your system - don't remove it!
Now to address the specific context of your situation:
That wiring configuration is indeed wrong and hazardous. The least dangerous part is using a single grounding screw to "tap" the green and white wires. This isn't correct. If nothing else was wrong with your pic, those wires should be tapped together with a pigtail going to the terminal. The worst part is the actual joining of the two systems together - at this location. That's just wrong.
From a logical standpoint, what would be accomplished in having two separate wires only to join them? From a safety standpoint, in an unbalanced circuit the neutral carries the unbalanced load back to the panel. So the "electrician" willfully energized the safety system in your home. It is okay to do this at the service/main because at this point they both have the same potential as the earth.
If you're in a home where the work in some or all areas is done this way, making it typical for your dwelling, call an electrician to devise a plan to fix it.
One thing to consider is if the bare ground goes back into the wall, it could be that they were adding a grounding system to the building. This is common in some locales as an accepted means to add a legitimate grounding system. Check with your building department before doing. The issue here though is, despite the respectable effort, the final connection is wrong. If this is the case, it might be safe to assume all or many of your outlets are this way.
What to do? Well, you have a few options.
If the grounding system in your pic is fine (per local building codes), but the connections are
wrong, you can simply separate the connection and use a grounded
outlet (NEC 406.4(D)(1)). I probably wouldn't recommend that unless you can, with
certainty, verify it's otherwise correct. The work you show would make any
self-respecting electrician cringe.
If you'd prefer to play it safe and would rather be inconvenienced with non-grounding adapters, you can simply exchange it out for a non-grounding receptacle (NEC 406.4(D)(2)(a)). Obviously you won't be using that ground any longer.
If neither of those are pleasing, you can opt to replace it with a GFCI provided you've labeled it "No Equipment Ground" (NEC 406.4(D)(2)(b)). You won't be using that ground in this situation either.