My main practice partner last Fall was a Link main. We went even the whole time. Peach is definitely one of Link's doable match ups. She doesn't have the tools that allow her to abuse Link's weaknesses, namely his slow speed and bad defensive and out of shield game. I think the match up is 6:4 at best for Peach.
Link's job in the match up is to camp you. Peach wins the close quarters fight. He has a sword so he can zone you decently well, but his moves have blind spots. Learn to exploit those blind spots. A big one is above and in front of him. He has no way to cover that quickly, safely, or easily. You can kind of fullhop float in front of him like you can against Ice Climbers. He has better ways than they to deal with it though, namely fullhop nair. However, fullhop nair is a big commitment by him. If he starts calling out your fullhop floats with his nair, then you can switch it up to make things difficult for him. Sometimes you can outspace the nair with a retreating fair. Otherwise, just mixup you float timings. Fullhop float, move in to bait the nair, fall back and down to the ground outside of his nair range and let him whiff the nair. You won't be able to get a direct punish, but he'll now be in a place he doesn't want to be. First of all, he has to come down. If he falls down with the nair, shield or CC it. If he double jumps out of the nair, follow his movement. How you punish exactly will depend on his habits and the stage.You may not be able to punish, but still shark under him with threats of attacks making him uncomfortable. If he starts outspacing your FC fairs, that means you can start being more aggressive with them and/or switching up your timings. Also try delaying them. The fullhop float fair threat is enough to make him act. Threaten the fair and respond to his response to your threat.
Dthrow dair doesn't work on Peach. You can
jump air dodge out at all relevant percents. It doesn't true combo until like 140%, especially if you DI behind her. However, at these percents where it is close to working, he can just uair you for the kill easily.
Don't be afraid to shield a lot. He has to go for a hard read to grab you and if he's wrong it means the start of a big combo or the start of an edge guard. Punish missed grab with uair or dash attack to combo, and fair/nair to edge guard. If he starts reading your shields, start using rolling away. He'll have to hard read that to punish if he even can.
Edgeguarding Link is difficult. He has so many options that it's hard or impossible to cover them all. I don't have anything to say except do standard Peach stuff. At the very least, try to edge guard him in a way where you can maintain stage control if he makes it back.
Don't try to out-camp him. It might be possible, but you're just playing to his strengths in the match up. This doesn't mean don't use turnips. Since he is so slow, he can't really punish you for pulling them. Pull one and use it to control his movement and/or start pressure.
Don't dsmash his shield unless he's facing away from you. Grab, jab, or fc nair after you land a good shield pressure FC aerial.
His boomerang can come out right away
Not really :/ It takes 30 frames to come out. Then it has to travel to you. Imagine if Marth's fsmash were 30 frames instead of its regular 11. You would totally be able to react to it every time! Shield or nair/bair the boomerang. Nair/bair will clang and you can act quickly.
Bomb pulls take 40 frames. Since he can pull them in the air, he can make it pretty safe, but it's still a long commitment. Use that time to move in and start at least zoning pressure. Shielding bombs is a good counter.
You don't get any true combos off uthrow.
Last but not least, stages. I think Peach's preferences go: FD > Yoshi's > Fountain > Pokemon = Dreamland > Battlefield. Avoid dreamland and battlefield because he can jump around on the platforms too well and camp you out. His camping is extremely strong on those two stages.
These are all just general nuances, but none of this is really going to give you a good edge, imo. The match up really comes down to fundamentals. If you could give more info about what you find difficult with the match up, I could talk through it more with you.
EDIT: The best thing you can do to abuse Link's low-tier-ness is to be extremely safe. Don't overextend at all. Threaten safely without high commitment. His options are generally bad. Push him to choose an option (not just a move, but any option -- jump, movement, shield, drift, and more -- and punish the option).