As shown in the other answers and comments this is a common issue that I've seen in several companies that I've worked in. Thinking it through, I suspect most companies struggle with the generic issue of allowing enough time for QA, testing, and automation once the feature is complete.
Generally, people may feel there is no clear guidance in Agile as to how to address this.
I would address this in two ways:
1) Testing happens before, during and after dev work. For example, if you practice BDD and write a failing test before the app code then you will be one step closer to your goal of keeping up.
2) A little discipline may be desired to allow more time for QA. For example, it's easy to say 'we will change to a process whereby dev works for a week and then QA has a week to test'. In reality, the work is usually not done in the first week and overflows into the second week leading to the same situation again. Try to address this with formal scheduled turnover and mileposts. For instance a calendar reminder "it's Friday, 3 pm. Is your code ready for testing?" You will also need to consider what would dev do for a week if no changes are allowed? Sitting idle for a week isn't going to work. This is a hard problem that is helped by a lot of exploring the issue and factors and by help from more senior folks who have experience in seeing the bigger picture and what would work best for the situation at hand.
In conclusion: You need to have detailed and difficult conversations with all the stakeholders in the development process in an open and caring environment that encourages all points of view in a non-threatening fun workplace where mistakes are just how people learn to do the right thing. In other words, A Good Culture.