The few problems that your team is facing, and the possible solutions:
Keeping up / too much load - Either the company is trying to roll out things too fast or your product does demand constant feature changes or the dev team does not do their part of testing. In either or a combination of these cases, a lot of load falls on the test team to keep up. This may result into tester's fatigue or the development of the attitude that even if we try our best, there is so much that there will always be bugs.
Automation - The test automation is as fragile as the test team that develops it. If you have engineers who can develop good test automation, that will help beyond words. If you are not confident enough, start with automating tasks that are relatively stable and most repetitive e.g. if you have to test 'create user' each test cycle, it is worth automating. Automation will also provide a level of creative satisfaction to the testers and something to look forward to. And it will, most certainly, give you more coverage at a greater speed.
Communicate with other members on the project - Communicate with the project manager or someone of the likes that the test team is being overloaded and if they can reconsider the number of features being rolled out each release. It is also highly likely that the development team feels the same. As a result, they might not be able to test their code thoroughly before passing it on to the test team. The company has a choice - add more features with more bugs, or less features with better quality.
Test team - Finally and most importantly (if this is not already done), have the manager meet each individual team member at least once in two weeks and get their feedback. How satisfied they are, what do they think about their work, load, etc. Get updates on what they did since the last meeting and what they are looking forward to achieve before the next meeting. Any suggestions to improve the quality of the product. This will also give an insight on how good the team actually is, are they motivated enough, if anyone is not doing enough, and much more. You can also do daily stand-ups for daily updates within the QA team.
Hope this provides some direction.