Sewing through thick layers of fabric can be a real challenge. When making bags often the trickiest parts to sew are straps and the side seams where you can end up with a lot of bulk.
Here are some tips for dealing with thick fabric/multiple layers:
1. Get rid of as much bulk as possible. For side seams I trim away as much as possible of any additional lining layers (in my patterns this is the "invisible lining"). You want to trim close to the stitching but still leave the stitches intact:
Then pressing--either with your iron or finger pressing--the side seams open flat will help immensely.
2. Use the hand wheel to sew across any thick areas. After breaking multiple needles attempting to sew through thick fabric I've gotten in the habit of sewing all my side seams and any areas with multiple layers of fabric using the hand wheel. So any time I'm attaching a flap or strap or sewing the Curvy Clutch wristlet strap I'm using the hand wheel.
3. Use a larger needle size. I switch up to a size 16 (100) needle once I get to the step of attaching the lining to the bag.