With a bit more time on my hands this morning,I dug out my Aldi 5 speed bench drill [been buried for the last year]
.Set it up on the bench and drilled a 5mm pilot hole,then reduced speed and went straight to 10mm with no problem. With a further speed reduction,11.5mm. Did not experience any real flexing using the 11.5,so I think there may be something not quite right with your drill,but do check your drill bits,no machine will cope with blunt drill bits.