"To stretch the truth" means to exaggerate or tell a story that is not completely truthful in order to make something seem more impressive or important than it really is. Full definition
Don't stretch the truth by expecting more than that. (novelinvestor.com)
Do not ever stretch the truth about what you have done on a resume. (job-hunt.org)
When it comes to online dating, many users will stretch the truth in the hopes of making themselves seem more appealing to other singles. (globaldatinginsights.com)