I am a Harry Potter fan, and I do not say that lightly. I have read the books 15-20 times EACH, and have seen the movies so many times that I can quote the lines word-for-word. I tried not to expect too much from the (book form of the) play Harry Potter and the Cursed Child, and honestly, I'm glad I didn't. Because if I had, I would have been let down. I have talked to numerous others that I know to be true Potter fans, and we all seem to be thinking along the same lines...it's not good, folks.
Some things (pretty much only criticisms) that Harry Potter fans are saying about Harry Potter and The Cursed Child: SPOILERS BELOW. READ AT YOUR OWN RISK.
1) I hated it.
2) It made me anxious.
3) They should have just quit while they were ahead.
4) I liked the 7th book's ending. Why couldn't they just leave it alone?
5) They waited too long to try to write this. Harry Potter lovers have already drawn their own conclusions about Harry's future life and family.
6) It reads like fan fiction.
7) It makes Harry look like a jerk.
8) It makes Harry's kid look like a jerk.
9) Why does it make Ron out to be an idiot?
10) At least we know Ron and Hermione are meant to be together, despite what JK said a few years back.
11) I don't accept the play as truth.
12) If Harry Potter's story continues, it should be in the same format and style and the original books.
13) Where did all the important characters go? Luna, the extended Weasley family, Teddy Lupin... Weren't they in Harry's life anymore?
14) There were plot holes.
15) The play didn't respect the original series in many regards.
16) It's too far-fetched.
17) Voldemort didn't reproduce. He just didn't.
18) I don't think Hermione would have wanted to be Minister of Magic.
19) Harry's son and Draco's son become best friends? I doubt that.
20) Why can't we hear more about James, Lily, and Hugo?
21) Why would a son of Harry Potter whose whole family was sorted into Gryffindor chose to be put in Slytherin?
22) Why dredge up the Cedric thing?
23) Weren't there more important deaths to be dredged up?
24) Is Albus supposed to resemble his namesake, Snape, when he goes through his "hating the world" phase?
25) How is the train Trolley Witch so old?
26) It's like the play was written to fulfill random fans' whims, hopes, and dreams.
30) Dumbledore's painting sure gets sentimental.
31) It didn't do any of the original characters justice.
32) The time-turner thing was already done in book 3.
33) Albus kisses his aunt...what the...
34) It was dumb.
35) It made me mad.
36) I kept hoping the end would make reading the whole thing worthwhile. It didn't.
37) There was no sense of adventure.
38) I was annoyed the whole time I was reading it.
39) I regret the purchase. It was expensive.
40) That was dumb.
41) Rose is horrible.
42) That was like a mash-up.
43) I don't know anyone who thought Harry's life/his kids' lives went down that way.
44) JK tried too hard to integrate something old with something new, and it didn't go over very well...
45) What happened to Hagrid?
46) I highly doubt Cedric would have become a Death Eater if he lost the Triwizard Tournament.
47) Umbridge definitely would have done well in Voldemort's regime. That's one thing they got right.
48) Why is there so much emphasis put on the awkwardness of Albus and Scorpius hugging?
49) There's not enough character development.
50) It was too short to cram all that stuff in there. They just tried too hard.
51) It's like they really wanted Voldemort to return, but ran out of ways to bring him back.
52) Why did Harry stay and watch his parents' murder?
53) It didn't give me that warm, fuzzy Hogwarts feeling.
So there you have it. It may seem critical, but Potterheads don't take additions to the Potter chronicles lightly.