Speed of sound in air in normal conditions is about 340m/s, and for an echo return that means 58.8us/cm, and 60us is a long time in electronics.
However you have to finish transmitting the pulse, shut down the transmitter, turn on/up the receiver, all in a millisecond or so. Ultrasonic transducers are resonant piezo plates - this means the transmitter will continue to oscillate for a long time if you don't damp it at the end of transmission - you should ensure the transmitter shuts down with the transmitter lines shorted to each other to reduce mechanical ringing.
Then you need to wait a little while before enabling the receiver (or you'll see the tail-end of the transmitting pulse at the receiver). So you basically ignore the received signal for a short delay (perhaps 1ms) and then look for the reflected pulse.
A high-performance sonar will increase the gain of the receiver as time goes by to compensate for the longer reflection path and provide a more constant threshold between signal and noise. Too much gain immediately after the ping goes out will mean the receiver is likely to falsely trigger on the tail-end of the outgoing ping (which will resonate both transmitter and receiver transducers remember). Perhaps you need to short out the receiver transducer till after the end of the transmitted pulse, to reduce its resonance until ready to receive reflected signal.
Separating the receiver from transmitter should reduce the ringing of the receiver during transmission, and the highly directional nature of ultrasound means that mounting them side-by-side some inches apart ought to be best.
To experiment further try placing the transmitter some distance from receiver and observe the transmitted pulse directly - this will mean you can experiment with receiver gain and so forth in a controlled environment