First, your family members may be eligible for Humanitarian Parole. Humanitarian Parole is designed to allow the applicants to expeditiously enter the United States due to a compelling emergency. The current processing time anywhere between 3 weeks to 3 months. To be eligible for Humanitarian Parole, you must demonstrate that your family members are either under a risk of harm or have some medical condition that needs immediate care or any other urgent humanitarian reason. Simply, showing the need to reunite your family is not enough. To file for Humanitarian Parole, you (i.e. the person in the US) would file form I-131 for each family member, attaching a detailed affidavit and evidence of the emergency condition. Each case is considered separately. However, it is frequently granted to family members of asylum applicants who are under risk of harm or need of urgent care.
Now, as to expediting the asylum interview or hearing, you must contact your local asylum office or file a motion with the immigration court. You can also request the expedited individual hearing at your master hearing. The judge will ask you why your hearing should be expedited, and if you demonstrate that your family is separated, the court may put your case on short list, which means that your hearing will be scheduled at the first available window, when, for example, another hearing gets cancelled or new time slot becomes available. Same applies for the USCIS asylum offices. If you demonstrate that your family is still in Iraq, the asylum office may put your case on a short-list of expedited cases.
Each asylum office had its own system of short-listing cases. For example, the Arlington Asylum Office, at least until recent times, was accepting short list requests for every case, as long as your application is full and complete and does not need any supplementation. In the past, the Arlington Asylum Office’s short list worked well, probably because not many applicants knew about it. Now, since almost every case gets short-listed, the likelihood of getting your case expedited without proof of exceptional reasons (e.g. family being separated) is much less. Therefore, it is imperative to provide evidence at the time when you request to short-list your case. To do so, you need to contact your local asylum office and following your local asylum office’s rules.