Having two kids has been so hectic specially during busy days where I can’t monitor their screen time. How do I manage it?
Welcome to the club! 
Honestly, on my craziest days, the tablet is basically a third parent. I try to set a loud kitchen timer. When it dings, the screens go in the jail (a high shelf) and they have to find something else to do. It’s not perfect but it keeps them from turning into zombies.
I just let them have at it sometimes. If I have a deadline and they are quiet, I count that as a win. 
You can’t do everything at once. Don’t beat yourself up over a few extra hours of Bluey.
I remember one Tuesday I had three back-to-back meetings. My 6-year-old spent four hours on a gaming app. When he finally looked up, his eyes were red and he couldn’t even tell me what he wanted for dinner. 
After that, I started pre-loading activities. I put out Legos or coloring books before the chaos starts so they have a physical backup when I can’t look at them.
@GorillaBlink the “screen jail” is a classic! We call ours the Forbidden Box. 
I find that if I give them a 5-minute warning before the timer, there are way fewer meltdowns. If I just grab the tablet, it’s game over for my eardrums.
I’m really worried about how much they are seeing when we aren’t looking. 
Even kids apps have weird stuff sometimes. I feel like if I’m not right there, I’m failing them. Does anyone else get that pit in their stomach when the house is too quiet?
@DroidPro, I totally get the guilt, but you are doing your best! Maybe try audiobooks? My kids will sit for an hour listening to a story. No blue light issues and it feels a bit more educational than random YouTube clips. 
My management style is just hiding the chargers. 
Can’t watch it if it’s dead! It’s a bit passive-aggressive, I know, but it saves me from the constant can I have five more minutes 100 times a day.
I’m so nervous they are going to get addicted or something. 
My oldest literally shakes when I take the phone away. Is that normal? I feel like I’ve lost all authority because I need the peace and quiet just to get through my chores.
Chill out, @RigidDatum. They aren’t going to break. My parents sat me in front of the TV for hours in the 90s and I turned out… mostly okay? 
Just make sure they have a payment system. 10 minutes of reading = 10 minutes of Minecraft.
@LinkRead hiding the chargers is genius. I usually just tell them the internet is sleeping when I need them to go outside. They haven’t figured out I’m lying yet, but I give it another six months before they realize I’m the one who flipped the switch. 