Use a trolley such as this one instead of carrying grocery bags. This shopping trolley is made exclusively from textile and it holds 43 litres.
From Amazon
ROLSER RD6 Logic RD6/MF, IMX092, 47.5 x 39.5 x 107 cm, 43 litres Maximum Load 40 kg, Marengo, 48,5 x 28,5 x 67 cm
Rolser
Rolser COM 8 BLACK TUBE MF Black - Black