Bugzilla extremely slow on EC2, RDS PostgreSQL

Question

The Problem

I have installed Bugzilla on an EC2 instance, using PostgreSQL RDS as my database. After entering a few dozen bugs, Bugzilla became unbelievable slow, and I get a lot of timeouts.

Server load

The machine is pretty idle:

14:26:00 up 23 days,  4:40,  2 users,  load average: 0.15, 0.16, 0.25
%Cpu(s):  0.3 us,  0.3 sy,  0.1 ni, 99.0 id,  0.0 wa,  0.0 hi,  0.0 si,  0.2 st

Log evidence

Example from Nginx log (hostname changed for security):

2014/07/16 14:18:32 [error] 6462#0: *5040 upstream timed out 
(110: Connection timed out) while reading response header from 
upstream, client: my.ip.my.ip, server: bugs.example.com, 
request: "POST /post_bug.cgi HTTP/1.1", 
upstream: "http://127.0.0.1:8081/post_bug.cgi", 
host: "bugs.example.com", referrer: "https://bugs.example.com/post_bug.cgi"

DB Analysis

Naturally, I suspected the database, so I run the following:

SELECT  datname,
    usename,
    now()-query_start AS duration,
    waiting,
    state,
    regexp_replace(query, E'[\\t\\n\\r\\s]+',' ', 'g' ) AS query_text
FROM pg_stat_activity
WHERE usename='bugs';

Which gave:

 datname | usename |    duration     | waiting | state |                   query_text
---------+---------+-----------------+---------+-------+------------------------------------------------
 bugs    | bugs    | 00:00:39.899945 | f       | idle  | SELECT value FROM bug_status WHERE is_open = 1

I think this is normal - the query is not waiting. When I executed the same query in a separate client, it returned immediately.

Cry for help...

Any idea why Bugzilla is so slow on my system?