Instead of concentrating on putting your weight on uke, concentrate on putting your weight into the mat. Keep your weight over your knees and elbows, and practice a bit of 'ju'. If uke bridges, give a little but keep your weight over your knees and elbows. If they rotate, move with them, keeping weight over your knees and elbows (or transition to a new hold). Instead of thinking of squashing uke, think of building a cage around them with your elbows and knees. Uke is free to move within the cage but don't let them escape it!
Looking at the practicalities, if your weight is on uke, they can feel your balance and exploit any lack of weight symmetry. Further, when they bridge, they get to use their whole bridge on you. These two things combined is probably why you are being reversed with them ended in yoko shiho gatame. By staying lighter on uke, they have less sensitivity over your position and balance. Also, when they bridge, the first half of their bridge is against air, so they are only using the final, decelerating portion of their bridge against you. It's a bit like comparing the force of a fully wound up punch with that of a half-jab. You are only letting them get off a half-jab.
At the end of the day, judo (as with life) is about maintaining your balance first and foremost. With that under control, you can turn your attention to others. This softer form of holding takes a bit to master. Give it a go but beware that your coach will probably give you a bollocking for not keeping weight on uke!