Changing the /etc/tinyproxy/tinyproxy.conf file section:

"# MaxClients: This is the absolute highest number of threads which will

# be created. In other words, only MaxClients number of clients can be

# connected at the same time.

#

MaxClients 100"

to:

MaxClients 1000

solved it for me. Did not change anything else. Mine is a dedicated VM, running as an LXC on Alpine Linux with CPU and 384MB of RAM