The full-size SD card module like the one linked by the OP is suitable for use only with 3.3V processors. While it has a 3.3V regulator to provide power to the SD card, the MOSI, SCK and CS lines coming in from the processor only have pullup resistors to 3.3V on the module, not level shifters. But those lines coming from a Mega would be 5V lines, which probably exceeds the absolute maximum voltage you can apply if the card is running at 3.3V. The 10K pullups would have no significant effect on the 5V level coming in. A possible fix is to insert diodes into the three lines so the processor can only bring the lines low, and the pullups bring them high. There would probably be some speed penalty with this fix because the pullups are 10K.
But I should say - some cards work just fine with the 5V lines. So this isn't necessarily the cause of the OPs problems.
