When the business keep piling on projects for the IT team, what generally happens is people are then split between multiple projects, with project managers generally being divided the most. The thinking is that a person can split themselves between multiple projects easily and that somehow they will manage to get all the projects done at the same time. The myth of multi-tasking has been dispelled many times so I’m not going to repeat that here except to say, don’t split people up between projects. Full stop. The biggest issue is lurking in this question is that fact that adequate portfolio-level prioritisation isn’t happening. The business needs to learn to prioritise properly. You can’t always just throw people at the problem, in fact, this often just makes things worse, as you end up with a bottleneck somewhere else, such as too many people working on the same code, or too many people trying to test in the same environment together.
IT and business need to work together to re-prioritise the projects so that the core team members can be fully allocated to a single project/product team. Once everyone is allocated, put all the other projects on hold until a team is freed up again. With the team focused on a single project, the throughput with increase and projects will start being delivered again.
In an enterprise you can have multiple project teams working at the same time, which is fine. However, you are generally constrained at some point because either all the teams may have to use the same environment to test or there may be some other infrastructural or process constraint. When this happens it doesn’t matter how many projects you’re running at the same time, your throughput will be limited by those constraints. Looking busy doing many projects isn’t solving your problem. Unblocking your constraint would be a much wiser investment.
Prioritisation is such a key part of agile. Work on one thing and get that thing finished.