r8168 by default just doesn't print rx errors. If you use ethtool msglvl to enable rx error logging, you should see the same errors with r8168. With r8168, even if it doesn't print rx errors, you should see the rx errors under /sys/class/net/enp4s0/statistics.

The status value indicates CRC errors. For whatever reason the checksum of received packets doesn't match. You could use wireshark or a similar tool to further analyze the issue. Or, to rule out a physical issue, try the easiest first: switch the cable.

Tried another network cable and the Rx errors persist.
With r8168 driver I have much less Rx errors. I got the outputs below
after ~4 hours using r8168 driver with the initial network cable.

$ sudo ethtool -S enp4s0
NIC statistics:
     tx_packets: 972174
     rx_packets: 1365445
     tx_errors: 23
     rx_errors: 5
     rx_missed: 0
     align_errors: 0
     tx_single_collisions: 0
     tx_multi_collisions: 0
     unicast: 1364929
     broadcast: 98
     multicast: 418
     tx_aborted: 0
     tx_underrun: 0

$ ifconfig enp4s0
enp4s0: flags=4163<UP,BROADCAST,RUNNING,MULTICAST>  mtu 1500
        inet 192.168.0.50  netmask 255.255.255.0  broadcast 192.168.0.255
        ether fc:aa:14:fc:4b:1b  txqueuelen 1000  (Ethernet)
        RX packets 1388034  bytes 1343088660 (1.2 GiB)
        RX errors 0  dropped 1  overruns 0  frame 0
        TX packets 984102  bytes 431299882 (411.3 MiB)
        TX errors 0  dropped 0 overruns 0  carrier 0  collisions 0
        device interrupt 30  base 0xb000

Weirdly the values of /sys/class/net/enp4s0/statistics/rx_errors
and /sys/class/net/enp4s0/statistics/rx_crc_errors remain 0 (zero).


I do not know anything about wireshark. I will read about it later.
Thanks.

I'm still having many Rx errors with r8169, now running kernel 5.16.1,
 and have no idea how to get helpful information with wireshark.
Sometimes even internet browsing is horrible and
Firefox takes ~15 seconds to fully load websites.
Right now I'm using r8168 with the same network cable and
rx error logging enabled with this command:

$ sudo ethtool -s enp4s0 msglvl rx_err on

$ sudo ethtool enp4s0
Settings for enp4s0:
	Supported ports: [ TP ]
	Supported link modes:   10baseT/Half 10baseT/Full
	                        100baseT/Half 100baseT/Full
	                        1000baseT/Full
	Supported pause frame use: Symmetric Receive-only
	Supports auto-negotiation: Yes
	Supported FEC modes: Not reported
	Advertised link modes:  10baseT/Half 10baseT/Full
	                        100baseT/Half 100baseT/Full
	                        1000baseT/Full
	Advertised pause frame use: Symmetric Receive-only
	Advertised auto-negotiation: Yes
	Advertised FEC modes: Not reported
	Link partner advertised link modes:  10baseT/Half 10baseT/Full
	                                     100baseT/Half 100baseT/Full
	                                     1000baseT/Half 1000baseT/Full
	Link partner advertised pause frame use: Symmetric Receive-only
	Link partner advertised auto-negotiation: Yes
	Link partner advertised FEC modes: Not reported
	Speed: 1000Mb/s
	Duplex: Full
	Auto-negotiation: on
	Port: Twisted Pair
	PHYAD: 0
	Transceiver: internal
	MDI-X: Unknown
	Supports Wake-on: pumbg
	Wake-on: g
        Current message level: 0x00000073 (115)
                               drv probe ifdown ifup rx_err
	Link detected: yes


dmesg log and the commands below do not report any Rx error.

$ ip -s -s link show enp4s0
2: enp4s0: <BROADCAST,MULTICAST,UP,LOWER_UP> mtu 1500 qdisc fq_codel state UP mode DEFAULT group default qlen 1000
    link/ether fc:aa:14:fc:4b:1b brd ff:ff:ff:ff:ff:ff
    RX:  bytes packets errors dropped  missed   mcast           
     516589996  467039      0       1       0     900 
    RX errors:  length    crc   frame    fifo overrun           
                     0      0       0       0       0 
    TX:  bytes packets errors dropped carrier collsns           
      28829051  282192      0       1       0       0 
    TX errors: aborted   fifo  window heartbt transns
                     0      0       0       0     104 

$ sudo ethtool -S enp4s0
NIC statistics:
     tx_packets: 285342
     rx_packets: 472224
     tx_errors: 60
     rx_errors: 0
     rx_missed: 0
     align_errors: 0
     tx_single_collisions: 0
     tx_multi_collisions: 0
     unicast: 471240
     broadcast: 79
     multicast: 905
     tx_aborted: 0
     tx_underrun: 0

$ ifconfig enp4s0
enp4s0: flags=4163<UP,BROADCAST,RUNNING,MULTICAST>  mtu 1500
        inet 192.168.0.50  netmask 255.255.255.0  broadcast 192.168.0.255
        ether fc:aa:14:fc:4b:1b  txqueuelen 1000  (Ethernet)
        RX packets 476660  bytes 525878530 (501.5 MiB)
        RX errors 0  dropped 1  overruns 0  frame 0
        TX packets 288003  bytes 29285844 (27.9 MiB)
        TX errors 0  dropped 1 overruns 0  carrier 0  collisions 0
        device interrupt 28  base 0x9000  

However, auto-negotiation does not work reliably with r8168.
Sometimes my network connects at 10 Mbps after login.
I'm seriously thinking to replace this realtek adapter
with an intel pci/pci-e adapter. I'm tired of so many problems with realtek.

The autonegotiation downshift indicates that the problem is on the physical side, e.g. cabling or RJ45 port problem on either side. There's no indication of a driver problem.

Could RAM failures cause RX errors? Memtest have detected RAM failures on my machine.
Another thing I have noticed is that the RX errors do not occur when the computer is idle,
even if there is network activity (by streaming audio over network, for example).

If your RAM is faulty, then best replace it and then re-check whether RX errors persist.

Check your power supply, if you have a weak power supply it can cause memory, CPU and network errors.

Closing this as probable hardware issues.

(In reply to Stephen Hemminger from comment #7)
> Check your power supply, if you have a weak power supply it can cause
> memory, CPU and network errors.

Power supply is good quality, manufactured by Seasonic, enough for my weak hardware. No rx error occurs when I connect a laptop and the affected computer directly, without my modem/router between them. I will ask my ISP provider to replace my modem/router. Thanks.

I just want to confirm that replacing the modem/router solved the RX errors.
Thank you all.