unity – Online multiplayer. Provoke lagged move on local player

I am developing an online multiplayer 2D videogame with Unity as a client and a server programmed in c ++.

I do the synchronization of movement through an interpolation so that the movement is smoothed, but I have a doubt that I cannot find an answer anywhere.

In any multiplayer game when the client has a bad connection (lag) the character moves awkwardly and not fluidly, but in reality I don’t understand why this behavior happens, currently in my game even though the client has a bad connection, it moves from fluid form because movement management is done locally and what is sent to the server is the player’s position.

How can I produce this in my game?

Thank you very much and sorry for my level of English. Thank you