You remove counters and put new ones on Galloping Lizrog all while its ability is resolving. Nothing can happen between the two, and no player may choose to take actions.

If Galloping Lizrog leaves the battlefield after its ability has triggered, you may remove any number of +1/+1 counters from among your creatures, even though you won’t put any on Galloping Lizrog.

If an effect causes Galloping Lizrog to enter the battlefield with +1/+1 counters on it, you may remove those counters with its triggered ability and put twice as many back on it.