  The local structure of correlated spin-orbit insulator Sr$_{2-x}$M$_x$IrO$_4$
(M = K, La) has been investigated by Ir L$_3$-edge extended x-ray absorption
fine structure measurements. The measurements were performed as a function of
temperature for different dopings induced by substitution of Sr with La or K.
It is found that Ir-O bonds have strong covalency and they hardly show any
change across the N\'eel temperature. In the studied doping range, neither Ir-O
bonds nor their dynamics, measured by their mean square relative displacements,
show any appreciable change upon carrier doping, indicating possibility of a
nanoscale phase separation in the doped system. On the other hand, there is a
large increase of the static disorder in Ir-Sr correlation, larger for K doping
than La doping. Similarities and differences with respect to the local lattice
displacements in cuprates are briefly discussed.
