Thank You all.
It makes the picture a bit clearer. There must be some strange idea about handshaking that has been mixed up with the TCP layer. What I notice in my experiment is that there will be no quick "SEND OK" unless something is sent back in response. If no data back the default delay seems to be near 300 ms.
To bring speed up I probably have to send a dummy response back from the server to the ESP8266 to help the sender get a "SEND OK" confirmation.
Since in my case the server is dedicated for the client it may solve the problem, but this is not the way it is supposed to work. Data is already sent - Even the reply is back before you get a "SEND OK" confirmation!
This can not be justified by buffer size. There is enough information in the TCP layer to figure out if data has been sent. There should not be any dependency on receiving incoming packages back. Those packages is another conversation. Sending should already be complete.
Sorry for "cross posting". Only to show real life scope signal images from ESP8266 strange "SEND OK" delay
where the delay is demonstrated.