My Court Jester had *just* turned 4 when we went and he was measured on every 40" ride ... and he made the cut every time, *just*.
Thank goodness, because he is pretty much a daredevil. He rode Soarin', Test Track, Splash Mountain, Big Thunder Mountain Railroad, Kali River Rapids & Goofy's Barnstormer. He was fine with most of the 3D stuff - loved Bug's Life & Muppets. We passed on Honey I Shrunk the Audience, he and I watched the fountains and played in the nearby exit pavilion with the music. Eeyore, Pirate Boy & Princess saw that one. We dragged him with the family on Haunted Mansion & Pirates of the Caribbean -- probably should have switched, because he didn't like them.
Clearly, he's a physical thrills kid. Every kid is different. I wouldn't want to have shown him too many YouTubes about rides, but every family is different. You might want to try that. Everyone taking a kid under 8 or so, or any sensitive child, should know that there is potential for "not fun" with plenty of attractions without height requirements.
My older kids were definitely taller earlier. He's a 50th percentile kid for height. My 7.5 year old probably would have made the 48" for Rock N Roller Coaster, but I didn't take her, since no one else was able or willing to ride that one. We did do Space Mountain, and she wasn't even asked to measure, if I recall correctly. My oldest is gigantic, at not much more than 10, he easily made the 52" cut off for the Tomorrowland Speedway. I think he's 54", he grew 2 3/4" last year and the pediatrician said to expect the same this year.

Court Jester won't be 52" when he's 12, most likely. Little dude's got style though.
