I have a some personal anecdotes from the past few years living in Los Angeles that might help explain the discrepancy between people's perceptions and the crime data.
A couple years ago two cars pulled up in front of the house across the street from mine, four people got out and started trying to break in by throwing a rock through the front window. The noise was loud enough to alert myself and a couple other neighbors on the street who came out to see the burglars fleeing back to their cars after the homeowner scared them away with a gun.
I called 911 and waited on hold for 2 hours for someone to answer only to tell me that the only the people who had their house broken into could file a report and to have them call back. When I went to tell this to the neighbor it turned out that he did try to report it but gave up after waiting on hold for an hour. In the end this home break in was never reported and did not make it into the crime data.
Last year, living in a different part of Los Angeles, I woke up at 4am to the sound of people yelling and bottles breaking outside. There was a group of very noticeably drunk people sitting on the hoods of their cars drinking in the street. I watched them long enough to determine they were all wasted, at least one of them would be driving, and if any of them drove they'd be a danger to themselves and others. Again I called 911, waited a half hour, and was told they'd dispatch someone. A half hour later they all got into separate cars and did indeed drive away, leaving empty beer bottles next to where their cars had been. It's hard to know if LAPD actually came to check because this all happened on the street adjacent to the back gate of the Hollywood police department and around 6am it gets a lot of traffic from police cars heading back to the station at the end of the shift. As far as I can tell this crime was never reported and did not get included in the data.
Also within the last year, on the same street adjacent to the police station, a homeless man seemingly in a manic episode was vandalizing parked cars while screaming about killing people of a certain protected group. This time I called 911, waited 20 minutes and told the operator what I saw. Because the homeless man had wandered to a different street and I couldn't see him anymore I was told it's not an ongoing crime and to call the LAPD non-emergency line so they can take the report. I called the non-emergency line and reached their voicemail. This is another crime that wasn't reported and didn't get included in the data.
The crime statistics are only as good as the data being collected. From personal experience I've witnessed 3 crimes which should have contributed to crime statistics but didn't. My suspicion is that cities across America are having similar problems simply taking in and documenting reports of crime. It could very well be that crime is actually rising, people are correctly perceiving that crime is rising, yet the data to substantiate the rising crime rate isn't getting recorded.