Rating moves on an overall basis is a mistake. There is no "third or so best ground move" in Peach's arsenal because they're all situational in how effective they are. That is why I admire Anti's MK because he didn't build it around abusing his "overall best moves", he built it on optimizing the most effective moves at the right times. Remember that.
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Ill, some advice I can give is to remain grounded more and focus on powershielding everything he throws at you excluding bombs (regular shield those instead). Practice on quickly catching bombs as they bounce off of your shield and use them as your own lead-in or retreat tools. Also remember that you have toad as an option if the Toon Link decides to come in and quickly catch the bomb in your face with a nair/fair.
Some other notes
- I noticed that Jerm started to gain momentum when he caught you in a rhythmic combo which was usually initiated by his zair. Don't jump if he is at about mid-range and already in an airdodge animation. Instead, either toad the zair if you know it's coming, powershield it (preferably), or shield to jab as he lands. Watch out for any surprise grab followups from him even if you're shielding; just be safe and roll either towards him or roll away (straight back into your shield) depending on his landing distance from you.
- Also be wary of his sh or full hop nair as a secondary option at that range. If he does it and hits your shield he will almost always pull back to avoid retaliation so don't be baited into doing any aerials OoS. Instead, you'll want to react accordingly depending on what type of nair he did...*points below*
Short hop nair: wait to see if he'll shoot a followup arrow or not, you can also follow up with dash attack OoS right before he lands. Time it precisely though.
Full hop nair: He has plenty of options from this position, so the last thing you want to do in this case is chase him into the air. Instead, use the moment to quickly pull a turnip and shield again right away. He'll have to make a choice on whether or not he wants to follow up with a drop down zair or stay in the air and retreat/pull a bomb. This is the situation you want to condition him into because at the very least: it resets both characters' positioning and you have a turnip in hand now.
Good luck on learning the matchup further, it's definitely one of Peach's trickier ones to deal with. Every motion and decision in this matchup counts because either chareacter can quickly snatch a lead.