Oh no, a Mummy rant!
My youngest just finished her (I)GCSEs, so I'm not toooo far distant from this. I worked, I breastfed, Hubby & I watched Christian evangelicals at 3am on workdays (foreign country, no grandparents, no babysitters, no other help) because Blessed Child #1 had chronic colic for months. I feel your pain. In no particular order:
(1) Back in our day (and before), chums, the cartoons were pretty suggestive as well, with Sylvester the Cat after all kinds of curvy female felines, not to mention Pepe LePeu (sp), so this is not a new problem.
(2) If you think video is bad, try buying clothes for a young girl without making her look like a slut.
(3) The real issue here is communication. Communicate with your child CONSTANTLY and you have a chance of dodging this mess, regardless of what they watch. Ours went through a very scary SlenderMan episode ::eye-roll:: not to mention the stuff on creepypasta, so you're not going to catch it all, but they need to know they can come to you with their problems, and they can't do that if you don't COMMUNICATE with them. Which means...
(4) Switch off the damned wireless throughout your house. Carry wired to your work machines and make it obvious that wireless is a privilege to be rationed out. Because...
(5) If the kids see inappropriate stuff at **home**, they're going to assume it's okay because YOU are the Home Authority. If they see it outside, there's no obvious parental imprimatur, so they're not going to be so receptive.
(6) Buy/download/save videos that you think are okay and have them on a USB stick that the kids can access while the wireless is OFF. Our kids learnt drawing, painting, yes even f**king opera with the wifi off. If you at least acknowledge their presence every now and then /sarc they will work with you and not think you're some irrelevant old fart.
Right now, your kids have the same degree of respect for you as you would have for a boss who sits you in front of a screen watching badly-written, derivative, angst-ridden, emotionally immature post-apocalyptic generational in-fighting films for hours every day. Without pay. Think about it.