Slow site

Please could anyone help me, I am at the end of my tether:

Our website has been live for just over 3 weeks. It is a Woocommerce site that sells giftboxes and has so far been a suucess. But there is one problem: Our SQL database keeps timing out and as a result makes the website slow. Even on a few occasions, we have reached our server limit.

We thought we’d cracked it yesterday by going through various plugins and deactivating them, but then our hosting support identified a serious backlog of MySQL queries on the server which would explain why the server was so slow.

An example of the error we are getting are on the root folder:

[ Moderator note: please wrap code in backticks or use the code button. Do not use blockquote. ]

[02-Apr-2014 21:31:21 UTC] WordPress database error Query execution was interrupted for query SELECT option_value FROM wp_options WHERE option_name = 'widget_tag_cloud' LIMIT 1 made by require_once('wp-load.php'), require_once('wp-config.php'), require_once('wp-settings.php'), do_action('init'), call_user_func_array, wp_widgets_init, do_action('widgets_init'), call_user_func_array, WP_Widget_Factory->_register_widgets, WP_Widget->_register, WP_Widget->get_settings, get_option

[02-Apr-2014 21:31:21 UTC] WordPress database error MySQL server has gone away for query SELECT option_value FROM wp_options WHERE option_name = 'widget_woocommerce_widget_cart' LIMIT 1 made by require_once('wp-load.php'), require_once('wp-config.php'), require_once('wp-settings.php'), do_action('init'), call_user_func_array, wp_widgets_init, do_action('widgets_init'), call_user_func_array, WP_Widget_Factory->_register_widgets, WP_Widget->_register, WP_Widget->get_settings, get_option

and admin:

[03-Apr-2014 05:34:15 UTC] WordPress database error MySQL server has gone away for query SELECT option_value FROM wp_options WHERE option_name = 'sbg_sidebars' LIMIT 1 made by require_once('wp-load.php'), require_once('wp-config.php'), require_once('wp-settings.php'), do_action('init'), call_user_func_array, sidebar_generator::init, sidebar_generator::get_sidebars, get_option
[03-Apr-2014 05:34:15 UTC] WordPress database error MySQL server has gone away for query SELECT term_id FROM wp_terms as t WHERE t.slug = 'bundle' made by do_action('admin_init'), call_user_func_array, WC_Bundles->woo_bundles_admin_init, term_exists

Please could anyone shed any light? I would be so grateful to receive any advice/feedback.

Thanks in advance.