Possible configure ESP8266 as wifi shield for Arduino?

Question

I know there's sketch to make ESP8266 becoming a simple web server. ESP8266 is cheap, small and consume less power compared to a Arduino Wifi shield.

Due to a more complex functionality requirement, I wish to place the web server code inside the Arduino itself (UNO, Nano or Pro Mini maybe). Is it possible to do this by configuring ESP8266 as a wifi bridge/shield to the Arduino? Or some mechanism methods which can route network traffics from ESP8266 to/from Arduino?

Updated
It connected to 4 relays & one I2C RTC DS1307.
It needs to sync RTC time with NTP server, and implement a web server which allows on-demand turn on/off each individual relay or by scheduling (e.g: turn on/off different relay at particular time of day). If RAM spaces allows, would like to implement a password access scheme for the web server as well

Answer 1

The ESP has 40 kb of ram and the average SPI flash rom used by ESP modules have 1 ~ 8 MB of code space. The average Arduino has much less of both. Uno R3 has the ATMega328, with 2kb of SRAM and 32kb of code space. Your trying to drive a truck with a mini cooper engine.

Answer 2

I can't comment on Passerby's answer so here is an answer to that question in the comments: What ESP8266 are you using? The simple ESP-01, the one with just 8 pins, does not have four I/O ports. However if you have a more advanced ESP8266(for example the ESP12 on a breakout board) you do have 4 I/O pins and you could just threw out the Arduino entirely. The ESP8266 is better suited to serve as a webserver than your arduino. So as Passerby suggested try just using your ESP8266 alone instead of combined with your Arduino. Have a look at this. It is a tutorial on how to build a webserver controlling two LED's. All you would need to do is replace the LED's with the relays and add another two outputs. Note that you do need another powersupply for your ESP8266 than the Arduino. The arduino can only supply about 40mA, while the peek current of an ESP8266 can be around 250mA. Good Luck!