My question isn't about whether or not the encryption on Bitlocker works, but whether or not there has been any security audits about potential backdoors Microsoft may have installed. I don't have a lot of trust that Microsoft wouldn't install deliberate backdoors into their software like this. I know I saw some reports that the government requested Microsoft add backdoors, but I am not sure if anything like that went forward.
Any time you install software from a vendor or project you are placing trust in that vendor or project not to have placed malicious code in there and also to have done a reasonable job of securing it.
Some people may suggest that open source is the answer to this, but without repeatable builds and a full source code audit done by someone you trust, there's less benefit to this than you might think to ordinary users.
Like most things in security this comes down to your threat model. If you don't trust Microsoft, then sure don't use their software. Bitlocker frankly is the least of your worries. if MS want to compromise you they can just trojan a security update (you do install security updates, right...).. Of course this is true whatever OS and application software you use. OSX, Apple can compromise you, Debian linux, the debian project can compromise you.
Realistically a better question is "is it likely that they want to"?
If you don't trust Microsoft, don't use Windows. Using Bitlocker doesn't make you more vulnerable to backdoors that Microsoft may have introduced. Cryptographic software is actually not the best place to put a backdoor: it has a fairly narrow job, it would be impossible to reliably hide what it's doing from someone running a debugger, and it would be rather cumbersome to extract the information that it stores. Sure, Bitlocker might use weaker encryption than what it claims, but someone who wanted your data would still need to have access to your machine to get at least the ciphertext.
If Microsoft wants to plant a backdoor, they can easily put one in the Windows kernel, which they made. It's hard to know what the kernel is doing: you can't trust a debugger that's running inside Windows, you'd have to run it in a virtual machine — and a stealthy backdoor might run differently or simply disable itself in a virtual machine (a lot of malware disables itself if it detects that it's running in a VM). If the system continuously exfiltrates information, even if that's done by the kernel, then it can be detected by network equipment. But if the kernel has a backdoor that makes it start exfiltrating only when it receives a certain signal, that's practically impossible to find out.
To put it another way, “is Bitlocker trustworthy?” is the wrong question. The right question is “is the trusted base of Bitlocker trustworthy?”, and the trusted base of Bitlocker includes a large part of Windows. Singling out Bitlocker doesn't make any sense. (Maybe suspecting Bitlocker is what “they” want you to do, so that you don't protect your data and so make “their” job easier!)
Is Bitlocker trustworthy?
No, because it does not provide a source code, therefore you cannot successfully verify if there are default system/admin ways to access it. Auditing it cannot fully cover all aspects, so therefore the audit is only valid form a functionality point of view.
There are alternatives that pre-date it and also offer the source code. If you really want something trustworthy, use one of these alternatives.
Additional note: given what Windows 10 is doing, there's a relatively small chance for Bitlocker not to have a way to be accessed by its designers.