Not being able to handle them being upset or crying is something it's worth working on reframing, as it can cause problems in a lot of different areas.
When they need to go through an experience which is going to be frightening or painful for them, such as an injection, they need you to be a calm, reassuring presence rather than seeing (or feeling/sensing) you being upset and worried by their own upset. They will get the sense "Something really bad must be happening - even mum is upset!" And it will exacerbate those feelings for them. Your role (ideally - I know I'm not always good at this!) is to be their rock in the storm of life. It can help to repeat a kind of mantra in your head which is something like "I know my child can handle this, I just need to be there for her while she goes through it."
When you hit toddler age and they are distraught because they want something they can't have, if you struggle with them being upset you will be tempted to give in to make them happy, especially in a situation where you think "Well... it wouldn't be that bad, and she really really wants it". This isn't very helpful because it makes it much more difficult to say no the times when you REALLY need to, which means you spend much more of your day in battles with your toddler, which is exhausting and stressful for everyone. I am not saying that you should never be flexible, but that specifically being uncomfortable with your child being upset about boundaries makes it much harder to have strong boundaries.
When they are upset because something bad happened (like they fell over, lost/broke a toy, somebody wouldn't play with them, etc) and your response is to try and fix it or distract them because you can't bear them being so sad, this does not allow them to work through and process the event. These things might seem minor to us but the act of being able to work through their sadness is an important part of learning to be OK with hard feelings. If they learn that sadness and anger should always be distracted or pushed away they are much more likely to turn to unhealthy coping mechanisms as teens/adults - drink, drugs, self-harm, overeating, lashing out at others, overspending etc. How do you tend to deal with feeling upset? How were your "scary" feelings handled as a child? It can be a new process to learn to just be there for your child rather than trying to make it better immediately. Or, you may have experienced the feelings being dismissed - "Stop crying, you're not a baby."/"It's OK, you're OK, it was only a little fall." Trying to fix or make better for our own kids can be a natural response to this, but still indicates discomfort around the expression of feelings. The most helpful approach long term is to understand, validate and be there for kids without trying to fix anything. They can handle it, given space and support to do so!
If you want to learn more about this way of parenting I'd recommend the books How To Talk So Little Kids Will Listen (And Listen So they will Talk), and The Whole Brain Child. Janet Lansbury also has a great blog/podcast which is free and she talks about accepting feelings a lot.