Relationship problems

Jess

So my bf has been messaging a girl on insta he used to date. I read his messages the other day and asked when the last time he talked to her was, and he said several months ago. That was the first lie. Then I asked If I could see his DMs and he said yeah but not at the moment. I looked at his phone later and the messages were deleted. He said he deleted them to avoid conversation....

Is he cheating?