Warning: there are spoilers in this article on past seasons of 'Grey's Anatomy'.
1. She is overdramatic, annoying, and narcissistic.
Literally everything she did, she dramatized. For example, she went after a cardio specialty and when she realized that Christina was better than her at it, she cried and gave up. She also routinely made everything about her. When Christina was left at the alter and Meredith had just broken up with Derek, she attempted to make everything about her because Callie was upset that Izzie and George slept together.
2. She breaks up George and Callie just to not be with George.
Izzie slept with George and then had the audacity to tell him that she is in love with him while he is married. Now, this also was George's fault, but George had feelings for Izzie before he got married and she turned him down. She also gave up on their relationship after he left Callie to be with her.
3. She leaves Alex to deal with all of her medical bills after she runs off.
Izzie got cancer and Alex stepped up to take care of her and help her through it. He even proposed to her and married her while she was receiving treatment. Once Izzy was in remission and cancer-free, she up and left Alex who had been there for her every step of the way and did everything she asked. She left him with all of her medical bills which left him drowning in debt and struggling.
4. She consistently strings Alex along and breaks his heart multiple times.
Throughout the entire time she was on 'Grey's Anatomy', she would lead Alex on and then turn around and be with another man. She did it with George after she told Alex she wasn't ready to be in a relationship, and she did it with Denny when Alex tried to help her because he loved her. All she ever did was break Alex's heart time and time again.
5. She has a romantic relationship with a patient and ends up essentially killing him.
Denny Duquette was a patient at Seattle Grace Hospital. Izzie was treating him, and she ended up falling in love with him. He needed a heart transplant and during the surgery, Izzie ends up cutting his LVAD wire and essentially ended up killing him. She faced little repercussions and in reality should have had her medical license revoked.
6. She has Alex’s kids and fails to ever tell him.
While we didn't actually see this occur and in reality, it was just a way for Alex Karev to exit the show, it still was a horrible thing for her to do. Izzie apparently had two children with Alex from the embryos she had taken and fertilized when she had cancer, never told him, and he had to find out after they were older.
7. She breaks up Alex and Jo when they are meant to be together.
After Alex and Jo had gone through so much and they were finally happy, Izzie had to go and break up yet another relationship. Izzie had Alex move in with her and their children and leave Jo. This is the third relationship that she broke up.
8. She was a bad doctor overall.
Essentially, all of her unprofessional choices really should have cost her her medical license. She always got overly involved with patients, showed her emotions far too much (for example, with the really old guy, she bantered and argued with him and called him rude names), or she was far too cold because she was trying to be like Christina. She always choked and created drama.
9. She could never accept that Meredith and Christina were better doctors.
She was always upset when Meredith or Christina got to do something that was a step ahead of her or when they would get the better surgeries because they were better doctors and lived for surgery, unlike her. She would always cry and whine when they got picked over her.
10. She would complain when Christina and Meredith would deal with their problems in their way without her help, and when they would try to help, she would still just complain. It was always ‘woe is me’ with her.
She was never satisfied. Meredith and Christina dealt with their problems in very similar ways and therefore did it together. She would always be upset that they wouldn't let her help because she was bright and sunny and they were dark and twisty. Even when they stepped in and helped her while they were dealing with their own problems, she would complain because it wasn't good enough for her like always.