Max, aged 19 from Birmingham and part of the elite Renault Sport Racing Academy, took the runner-up spot in the first of two races at Germany’s Nürburgring circuit.
He followed that up with a dominant victory in race two to make it four podium finishes in a row in his R-ace GP team prepared car.
The results had moved him to within just 13.5 points of championship leader, Denmark’s Christian Lundgaard, ahead of the Eurocup’s next two rounds – set to take place as we went to press at another German circuit, Hockenheim (22-23 Sept).
After that just the final two rounds remain – at the Barcelona Grand Prix circuit (20-21 October).
Max said: “This weekend, we were fast from the beginning. We kept our heads down over the three days and our work paid off.
“Before Budapest, I thought that it would be difficult to compete for the title, but we’re right back in the game! We have always known that we have the potential, but we had to get rid of a bit of misfortunes.
“I hope that we can keep the momentum going at the last two meetings and I am really looking forward to see how it all plays out!”
Max is hoping to become the latest Briton to clinch the prestigious Eurocup crown. In 2015 it was won by Jack Aitken – now official reserve driver for the Renault Sport Formula One Team – and in 2016 Llando Norris who it has been confirmed will make his Formula 1 debut in 2019 with the Renault-powered McLaren team.