From watching this video I have some feedback for you. I'm seeing an overreliance on attacks that aren't safe.
-The dash attacks are probably connected to your earlier complaints about being grabbed. On shield, that dash attack is a free grab.
-There are a few times where you hover above the Link, then use a fair and get punished. From my matchups against Peach players, I find that her dair is hard to punish when she's floating above me, making it safe shield pressure, and damage if I drop my shield. Do you have your controls set up to allow you to do multiple attacks while continuing to float (as in, while holding a jump button)?
-I saw your comment about hovering with Links, nevertheless I'd like to have seen you hover more. I think hovered dairs or running grabs would have been a more successful successful approach than a dash attack nearly every time. Remember, you don't have to go straight for him. Say you went for a hovering dair approach - Link's usmash is a powerful move and has great priority. I'm sure Peach's dair loses to it. However, if you hover just above him and out of range, you can bait out that attack and punish it afterwards with a hovering dair or some other attack. Peach's air spacing is similar to Jigglypuff's in that way.
-I notice that you were unwilling to chase Link offstage. Mostly what I saw were turnips being thrown in his general direction. The problem with the turnips is, you were throwing them towards him from onstage, so due to the trajectory of the throws they were either passing safely over him, or through him during his ledge invincibility. This is an attempt at edge-guarding, but not a very proactive one. Without going underneath the stage to pressure his recovery directly, there are a few things you can do that put you at minimal risk. Grab a turnip, hover over the stage and throw it downwards (or z-drop, or toss it up, if there's time for it to reach your opponent). If they're recovering to the ledge from above, you can hover out meet them. Use a faster, safer attack like nair or multihit one like dair to attempt gimps. Peach has no problem moving horizontally, and the further offstage you go to meet someone, the less likely they'll have time to throw out an attack. But as you know, unless you get spiked, Peach's up b can get you back to the ledge easily. Not so for your opponent. Now, I could be wrong about my aerial attack recommendations (which ones to throw out etc), but you should get comfortable offstage. Peach is a character that can go deep very safely. Keep her 2nd jump and up b and she can recover from practically anywhere. That is one of her strengths. Offstage game is huge in Smash. I feel alright against some of the better players here when onstage, I can generally keep up and the battle isn't too one sided. But when I get offstage is when the match starts getting stacked against me. I can fight Karsticles' customs Ganondorf somewhat evenly onstage, but if he spikes me with a dair as I'm recovering the battle goes from even to one sided. Peach doesn't have a spike, but she can go so far out to disrupt recoveries that she doesn't need one.
-I already mentioned it a bit, but knowing your character's strengths and minimizing their weaknesses is important. If an attack leaves you vulnerable, a) is there a safer option that does the same thing, and b) if it's unsafe, use it rarely enough that the opponent doesn't expect it.
Keep it up!
This is good advice against many characters, but I don't think Link is one of them. The one in the video was pretty aggressive and didn't hang back to use projectiles, but Link is one of the last characters that would need to go towards their opponent, besides to hit the killing blow. Many Links on FG are willing to keep peppering people with bombs and boomerangs past 150%, and their projectiles have more range and speed than turnips.
