Every time I have to leave the house for appointments or whatever my 2 youngest kids throw an absolute fit. They’re 4 and 2. No matter if I leave them with dad, grandma, whoever. The 2 year old, I kinda get. The 4 year old throws me for a loop. I know they’re not being mistreated and they always have fun when I’m gone, but getting out of the door is so hard. It’s like most of the time I have to sneak out while they’re occupied to avoid clinging to my legs and screaming
You’re they’re safe space and it’s very normal.
It just sounds like they have separation anxiety. They’re used to being right there with you. Just remind them that you’ll come back and it won’t be very long. Give them a quick hug and kiss go do what you need and then when you come back hug them and say see mommy comes back. That way they’ll get used to knowing that you are coming back.
Dealing with this with my son also, he’s 5.
It’s so hard at this age but normal. What helps with my oldest, is I give him a timeframe… “I’ll be back by snack time.” That seems to ease his anxiety because he knows when I will come home. My youngest we have to distract for the most part. You could also try making a list of things to do while you’re gone, like build a fort, play with dolls, etc, so the oldest has a goal and their own set of distractions.
Enjoy it! There will be a day, usually the teen years, they will want you to leave! 
Just keep going out and they’ll eventually get used to it. It’s hard at first for moms too, I know but it’s good for them and you to be away from each other, so just keep going out, also don’t sneak out. Keep up the process of loving goodbyes, they’ll be fine!
Aw…they just loooove their mom! Do try to “prep” them several hours in advance and let them know when you’ll return. Time goes fast—before you know it they’ll be teenagers sneaking out the window at 2am away from your watchful eyes!
You should try letting them know your leaving instead of sneaking out. Say “mommy is leaving now BUT I will be back”. Make the good bye quick and leave. Don’t draw it out. Do it consistently if you can
I wouldn’t leave if my kids did that. First time I was ever away from my kids were their first day of kindergarten.
Never never ever sneak out,that just freaks them,one minute your there the next your gone,and then they go searching…once you leave they’re are fine,experienced great grandma here
Normal kid stuff. My 8 year old still don’t like when I do things lol
They can smell fear and dread. Communicate with them , tell them when your leaving and why. Explain who will be there for them. Prep them a day ahead of time. Say goodbye ,give hugs and go!
You tell them you’re going.
Tell them there will be small rewards for a good behavior report.
You say your goodbyes ONE TIME and GO.
Ignore the wailing and crying and such.
Yes, it will feel like it’s killing you.
Put your music on & wear your air pods if necessary, before saying goodbye and crank it up after that .
Then you come back bring a small reward; idk, stickers, donut holes, whatever their thing is.
Lather rinse repeat.
I know it cuts you up; you’re training them in “object permanence”, reassuring them you always come back, and reinforcing better behavior.
If they are old enough, create an easy to read sheet outlining what good behaviors will earn them a treat. Post at eye level somewhere they can see.