Can one cut and bulk simultaneously?

I’ve googled this lots of times but all the information out there is very confusing and conflicting! The reason I’m curious about it is because I always find myself conflicted about which of my goals I should focus on when training and picking program etc I really want to build muscle but then I somehow feel that my muscles would be more visible if I lost fat (not weight), haha but then I get sad when I stop heavy weight training because somewhere in my messed up brain I think “loosing muscle“ equates to loosing all the gains I’ve made in terms of fitness blah blah blah. I’m working on fixing that attitude DW!

Anyways despite my weird reasonings, is it possible to gain muscle mass and loose fat at the same time? How do you suggest one eats during this process?and which program best suits? Thanks x